
\(\omega \)  and \(\lambda \)  are real. Then \(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre}) = \mathbb {R}\) .

\(|\mathrm {Im}(\omega )| < |\mathrm {Im}(\lambda )|\) . Then \(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre}) = {- \mathrm {sgn}(\mathrm {Im}(\lambda ))}\) .

\(|\mathrm {Im}(\omega )| = |\mathrm {Im}(\lambda )| \ne 0\) . Then \(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre}) = {- \mathrm {sgn}(\mathrm {Im}(\lambda ))} \cup \lbrace  0 \rbrace \) .

\(|\mathrm {Im}(\omega )| > |\mathrm {Im}(\lambda )|\) . Then \(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre}) = .\) 
The same is true with \(D_{\omega , \lambda }^\mathrm {pre}\)  replaced by \(D_{\omega , \lambda }^{\min }\)  throughout.
Integrating by parts we find that for \(f = \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} \in C_\mathrm {c}^{\infty }(\mathbb {R}_+,2)\)  we have
\(\mathrm {Im}( f| D_{\omega , \lambda } f ) = - \mathrm {Im}(\lambda + \omega ) \int _0^{\infty } \frac{|f_1(x)|^2}{x} x̥ - \mathrm {Im}(\lambda - \omega ) \int _0^{\infty } \frac{|f_2(x)|^2}{x} x̥ .\) 
In the four cases listed in the proposition we have: both terms are zero in Case \(1.\) , both terms are nonzero (except for \(f=0\) ) and have the same sign as \(- \mathrm {Im}(\lambda )\)  in Case \(2.\) , one term is zero and the other has the same sign as \(- \mathrm {Im}(\lambda )\)  in Case \(3.\)  and the two terms have opposite signs in the last case. Therefore inclusions of numerical ranges in the specified sets are clear, except for the third case. Then in order for \(\mathrm {Im}( f| D_{\omega , \lambda } f )\)  to vanish, one of the two \(f_j\)  has to be zero. It is easy to check that this implies \(( f| D_{\omega , \lambda } f ) =0\)  (but not \(f =0\) ).
We have to show that the obtained inclusions are saturated. The
homogeneity of \(D_{\omega , \lambda }^\mathrm {pre}\)  implies that
\(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre})\)  is a convex cone. Thus to establish
the result in Case \(1.\)  it is sufficient to show that both signs of \((f | D_{\omega , \lambda } f )\)  are possible. We choose a nonzero \(\varphi \in C_c^{\infty }(\mathbb {R}_+,2)\)  with \(\Vert  \varphi \Vert _{H_0^1}=1\)  and put \(f_{\pm , t}(x) = \begin{bmatrix} \varphi (x-t) \\ \pm \varphi ^{\prime }(x-t) \end{bmatrix}\)  for \(t \ge 0\) . Then \(\Vert  f_{\pm , t} \Vert _{L^2}=1\)  and
\(( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} ) = \pm 2 \int _0^{\infty } |\varphi ^{\prime }(x)|^2 x̥ - \int _0^{\infty } \frac{1}{x+t} \left( (\lambda +\omega ) |\varphi (x)|^2 + (\lambda - \omega ) |\varphi ^{\prime }(x)|^2 \right) x̥ .\) 
The first term is nonzero, has sign \(\pm \)  and does not depend on \(t\) , while the other converges to zero for \(t \rightarrow \infty \) . Therefore \(\pm ( f_{\pm , t} |D_{\omega , \lambda } f_{\pm , t} ) \ge c_{\pm } >0\)  for large enough \(t\) .
Next we suppose that \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )| \ne 0\) . It
is sufficient to show that \(-\)  is included in the numerical range
for \(\mathrm {Im}(\lambda )<0\) . Arguing as below (REF ), we deduce
that there exist constants \(t_0 > 0\)  and \(c_{\pm } >0\)  such that \(\pm \mathrm {Re}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} ) \ge c_{\pm }\) 
for \(t \ge t_0\) . Let \(\delta = \mathrm {Im}\, (f_{ \pm , t_0}|D_{\omega ,\lambda } f_{ \pm , t_0} )\) . Then \(\delta >0 \) . The function \(t \mapsto \mathrm {Im}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} )\)  is continuous and converges to zero for \(t \rightarrow \infty \) , so for every \(\epsilon \in ]0, \delta ]\)  there exists \(t \ge t_0\)  such that \(\mathrm {Im}\, ( f_{\pm , t} | D_{\omega ,\lambda } f_{\pm , t} ) = \epsilon \) . By convexity of numerical ranges this implies \([-c_- + \mathrm {i}\epsilon , c_+ + \mathrm {i}\epsilon ] \subset \mathrm {Num}(D_{\omega ,\lambda })\) . Homogeneity implies that for every \(s>0\)  we have \(\left[ -\frac{c_- s}{\epsilon } + \mathrm {i}s, \frac{c_+ s}{\epsilon } + \mathrm {i}s \right] \subset \mathrm {Num}(D_{\omega , \lambda })\) . Every \(k\)  with \(\mathrm {Im}(k)=s\)  is in this interval for small enough \(\epsilon \) .
Similar argument shows that in Case \(4.\)  there exist \(c_{\pm } >0\)  and \(\delta >0\)  such that for every \(\epsilon \in ] 0 , \delta ]\)  there exist \(g_{\pm ,\epsilon } \in C_c^{\infty }(\mathbb {R}_+,2)\)  with \(\Vert  g_{\pm ,\epsilon } \Vert _{L^2}=1\) , \(\pm \mathrm {Re}( g_{\pm ,\epsilon }|D_{\omega , \lambda } g_{\pm ,\epsilon } ) \ge c_{\pm }\)  and \(\left| \mathrm {Im}( g_{\pm ,\epsilon }| D_{\omega , \lambda } g_{\pm ,\epsilon } )\right| \le \epsilon \) . On the other hand for nonzero \(f \in C_c^{\infty }(\mathbb {R}_+,2)\)  with \(f_1=0\)  or \(f_2=0\)  we have that \(( f| D_{\omega , \lambda } f )\)  is proportional to \(\omega -\lambda \)  or \(- \omega - \lambda \) , respectively, with a positive proportionality constant. Using homogeneity we can even construct functions \(f\)  with the proportionality constant equal to 1 and \(\Vert  f \Vert =1\) . Next we observe that if \(\epsilon \)  is taken to be sufficiently small, the convex hull of \(( g_{+, \epsilon } | D_{\omega , \lambda } g_{+, \epsilon } )\) , \(( g_{-, \epsilon } | D_{\omega , \lambda } g_{-, \epsilon } )\) , \(\omega - \lambda \)  and \(- \omega - \lambda \)  contains zero in its interior. Therefore the smallest convex cone containing it is the whole \(.\) To prove the last statement, first note that \(\mathrm {Num}(D_{\omega , \lambda }^{\min })\)  is contained in the closure of \(\mathrm {Num}(D_{\omega , \lambda }^\mathrm {pre})\) . Therefore in Cases \(1.\)  and \(4.\)  there is nothing to prove. We consider Case \(2.\)  We have to show that if \(g \in \mathrm {Dom}(D_{\omega , \lambda }^{\min })\)  is such that \(\mathrm {Im}( g| D_{\omega , \lambda } g ) =0\) , then \(g=0\) . We choose \(\epsilon >0\)  and \(f \in C_c^{\infty }(\mathbb {R}_+,2)\)  such that \(\Vert  f - g \Vert _{\mathrm {Dom}(D_{\omega , \lambda }^{\min })} < \epsilon \) . Then
\(\mathrm {Im}( f| D_{\omega , \lambda } f ) = \mathrm {Im}\left( ( g | D_{\omega , \lambda }^{\min } (f-g) ) + ( f-g | D_{\omega , \lambda }^{\min } g ) + (f-g | D_{\omega , \lambda }^{\min } (f-g) ) \right),\) 
so \(|\mathrm {Im}( f| D_{\omega , \lambda } f )| \le 2 \epsilon \Vert  g \Vert _{\mathrm {Dom}(D_{\omega , \lambda }^{\min })} + \epsilon ^2\) . On the other hand for any \(t >0\)  we have
\(& |\mathrm {Im}( f| D_{\omega , \lambda } f )| \ge \frac{|\mathrm {Im}(\lambda +\omega )|}{t} \int _0^t |f_1(x)|^2 x̥ + \frac{|\mathrm {Im}(\lambda -\omega )|}{t} \int _0^t |f_2(x)|^2 x̥  \\& \ge \frac{|\mathrm {Im}(\lambda +\omega )|}{t} \left( \int _0^t |g_1(x)|^2 x̥ - 2 \epsilon \Vert  g \Vert ^2 \right) + \frac{|\mathrm {Im}(\lambda -\omega )|}{t} \left( \int _0^t |g_2(x)|^2 x̥ - 2 \epsilon \Vert  g \Vert ^2 \right). \nonumber \) 


Comparing the two derived inequalities and taking \(\epsilon \rightarrow 0\)  we find that
\(\int _0^t |g_1(x)|^2 x̥ = \int _0^t |g_2(x)|^2 x̥ = 0.\) 
Since \(t\)  was arbitrary, \(g=0\) . Case \(3.\)  may be handled analogously.
It is convenient to describe the numerical ranges of operators \(D_p\) 
in terms of \([a:b] \) . It can be related to parameters \(\omega ,\lambda , \mu \)  by recalling that \([a:b] = [- \mu : \omega + \lambda ]\) 
if \(\omega +\lambda \ne 0\)  and \([a:b] = [\omega - \lambda : - \mu ]\) 
if \(\omega - \lambda \ne 0\) . No such expression exists on the zero
fiber. We will also choose a representative \((a,b) \in [a:b]\) . We note
that the condition \(\mathrm {Im}(\overline{b} a) =0\)  is equivalent to the existence of a real representative \((a,b)\) , which is also equivalent to the statement that \([a:b]\)  belongs to the real projective line \(\mathbb {RP}^1\) . If \([a:b] \notin \mathbb {RP}^1\) , then \(\mathrm {sgn}(\mathrm {Im}(\overline{b} a)) = \mathrm {sgn}\left( \mathrm {Im}\left( \frac{a}{b} \right) \right)\) .
Proposition 37 The numerical range of \(D_p\)  may be characterized as follows.

If \(\omega , \lambda \in \mathbb {R}\)  and \([a:b] \notin \mathbb {RP}^1\) , then \(\mathrm {Num}(D_p) = \mathbb {R}\cup {- \mathrm {Im}\left( \overline{b} a \right)}\) .

If \(\mathrm {Re}(\mu )=0\)  and \(\mathrm {Im}(\overline{b} a) \mathrm {Im}(\lambda )<0\) , then \(\mathrm {Num}(D_p) =.\item If \) Re()<0\( and \) [a:b] RP1\(, then \) Num(Dp) = .

In every other case \(\mathrm {Num}(D_p) = \mathrm {Num}(D_{\omega , \lambda }^{\min })\) .


If \(p = \overline{p}\) , then \(D_p\)  is self-adjoint, so \(\mathrm {Num}(D_p) \subset \mathbb {R}= \mathrm {Num}(D_{\omega , \lambda }^{\min }) \subset \mathrm {Num}(D_p)\) . If \(|\mathrm {Im}(\omega )| > |\mathrm {Im}(\lambda )|\) , then \( \mathrm {Num}(D_{\omega , \lambda }^{\min }) \subset \mathrm {Num}(D_{p})\) .
Let \(\eta (x) = x^{\mu } \begin{bmatrix} a \\ b \end{bmatrix}\)  and consider \(f = \begin{bmatrix} f_1 \\ f_2 \end{bmatrix} \in C_c^{\infty }(\mathbb {R}_+,2) + \mathrm {span} \lbrace  \chi \eta \rbrace \) . Then
\(\mathrm {Im}( f | D_{p} f ) = \mathrm {Im}\int _{0}^{\infty } \left[ \frac{{x̥} \left( \overline{f_2(x)} f_1(x) \right) - \frac{(\lambda + \omega ) |f_1(x)|^2 + (\lambda - \omega ) |f_2(x)|^2}{x}  x̥.}{}\right.By construction, there exist x_0 >0 and c \in  such that for x < x_0 we have f(x) = c \, \eta (x), and hence \overline{f_2(x)} f_1(x) = \mathrm {Im}(\overline{b} a) x^{2 \mathrm {Re}(\mu )}. If \mathrm {Re}(\mu )>0 or \mathrm {Im}(\overline{b} a)=0 (which is equivalent to [a:b] \in \mathbb {RP}^1 \subset \mathbb {CP}^1), then \overline{f_2(x)} f_1(x) vanishes for x sufficiently large and for x \rightarrow 0. Therefore \int _{0}^{\infty } \frac{{x̥} \left( \overline{f_2(x)} f_1(x) \right) x̥ =0 and the proof goes as for Proposition \ref {num_range_minimal}.}{W}e consider the case \mathrm {Re}(\mu ) = 0 and \mathrm {Im}(\overline{b} a) \ne 0. Then\begin{equation}\mathrm {Im}( f | D_p f ) = - |c|^2 \mathrm {Im}(\overline{b} a) - \mathrm {Im}\int _{0}^{\infty } \frac{(\lambda + \omega ) |f_1(x)|^2 + (\lambda - \omega ) |f_2(x)|^2}{x} x̥.\end{equation}If \omega , \lambda \in \mathbb {R}, then \mathrm {Im}( f | D_p f ) = - |c|^2 \mathrm {Im}(\overline{b} a) and we have \mathbb {R}= \mathrm {Num}(D_{\omega , \lambda }^{\min }) \subset \mathrm {Num}(D_p), so~\mathrm {Num}(D_p) = \lbrace  k \in  | \, \mathrm {Im}(\overline{b} a) \mathrm {Im}(k) \le 0 \rbrace . In the case |\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )| \ne 0 there are two possibilities. If \mathrm {Im}(\overline{b} a) \mathrm {Im}(\lambda )>0, then all terms in (\ref {eq:im_mu_im_fhf}) have the same sign and one has \mathrm {Num}(D_p) = \mathrm {Num}(D_{\omega , \lambda }^{\min }). Otherwise \mathrm {Num}(D_p) = . Indeed, consider f = \frac{\chi \eta }{\Vert  \chi \eta \Vert } with shrinking support of \chi \ge 0. A simple calculation shows that for these functions the integrand in (\ref {eq:im_mu_im_fhf}) vanishes, while the first term grows without bound.\) 
Next, we suppose that \(\mathrm {Re}(\mu ) <0\) , \(\mathrm {Im}(\overline{b} a) \ne 0\) . Put \(f = \varphi \eta \)  with \(\varphi \in C^{\infty }([0 ,\infty [, \mathbb {R})\)  vanishing exponentially at infinity. Then \(f \in \mathrm {Dom}(D_p)\)  and \((D_p f)(x) = \varphi ^{\prime }(x) \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \eta (x)\) . Thus
\(( f | D_p f ) = 2 \mathrm {i}\, \mathrm {Im}(\overline{b} a) \int _0^{\infty } \varphi (x) \varphi ^{\prime }(x) x^{2 \mathrm {Re}(\mu )} x̥.\) 
If \(\varphi \ne 0\)  vanishes at zero, the integral is positive, as can be seen by integrating by parts:
\(\int _0^{\infty } \varphi (x) \varphi ^{\prime }(x) x^{2 \mathrm {Re}(\mu )} x̥ = \frac{1}{2} \int _0^{\infty } \frac{{x̥} (\varphi (x)^2) x^{2 \mathrm {Re}(\mu )} x̥ = - \mathrm {Re}(\mu ) \int _0^{\infty } \varphi (x)^2 x^{2 \mathrm {Re}(\mu )-1} x̥ > 0.}{O}n the other hand, for \varphi (x) = \mathrm {e}^{- \frac{x}{2}} the integral is negative:\begin{equation}\int _0^{\infty } \varphi (x) \varphi ^{\prime }(x) x^{2 \mathrm {Re}(\mu )} x̥ = - \frac{\Gamma (2 \mathrm {Re}(\mu )+1)}{2} <0.\end{equation}By Proposition \ref {num_range_minimal} and the fact that \mathrm {Num}(D_p) is a convex cone, we have \mathrm {Num}(D_p) = .\) 
We adopt the convention saying that operators with the numerical range
contained in the closed upper half-plane are called dissipative. Dissipative operators which are not properly contained
in another dissipative operator are said to be maximally
dissipative. This condition is equivalent to the inclusion of the spectrum in the closed upper half plane. Maximally dissipative operators may also be characterized as operators \(D\)  such that \(\mathrm {i}D\)  is the generator of a semigroup of contractions.
Corollary 38 \(\pm D_p\)  is a dissipative operator if and only if one of the following (mutually exclusive) statements holds:

\(\omega , \lambda \in \mathbb {R}\)  and \(\mp \mathrm {Im}(\overline{b} a) \ge 0\) .

\(\pm \mathrm {Im}(\lambda )<0\) , \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )|\)  and \(\mathrm {Re}(\mu ) >0\) .

\(\pm \mathrm {Im}(\lambda )<0\) , \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )|\) , \(\mathrm {Re}(\mu ) =0\)  and \(\pm \mathrm {Im}(\overline{b} a) \le 0\) .

\(\pm \mathrm {Im}(\lambda )<0\) , \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )|\) , \(\mathrm {Re}(\mu )<0\)  and
\(\mathrm {Im}(\overline{b} a) =0\) .

Furthermore, if these conditions are satisfied then \(\pm D_p\)  is maximally dissipative.

Corollary 39 Let \(\omega , \lambda \)  be such that \(\pm D_{\omega , \lambda }^{\min }\)  is dissipative, i.e. \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )|\) , \(\pm \mathrm {Im}(\lambda ) \le 0\) . There exists \(p \in \mathcal {M}_{- \frac{1}{2}}\)  such that \(D_{\omega , \lambda }^{\min } \subset D_p\)  and \(\pm D_p\)  is maximally dissipative. In particular \(\pm D_{\omega , \lambda }^{\min }\)  admits a maximally dissipative extension which is homogeneous and contained in \(\pm D_{\omega , \lambda }^{\max }\) .

We present the proof for the upper sign. The other part of the statement then follows by taking complex conjugates. If \(\omega , \lambda \in \mathbb {R}\) , it is possible to choose \(p\)  with \(\mp \mathrm {Im}(\overline{b} a) \ge 0\) . Now let \(\mathrm {Im}(\lambda ) <0\) , \(|\mathrm {Im}(\omega )| \le |\mathrm {Im}(\lambda )|\) . If \(\omega ^2 - \lambda ^2 \notin ] - \infty , 0 ]\) , we can choose \(\mu \)  with \(\mathrm {Re}(\mu ) >0\) .
Next suppose that \(\omega ^2 - \lambda ^2 \le 0\) . If the inequality is strict, then there exist two possible choices of \(\mu \)  differing by a sign, so the condition \(\mathrm {Im}(\overline{b} a) \le 0\)  is satisfied for at least one choice. If \(\omega ^2 - \lambda ^2 =0\) , then either \(\omega + \lambda \)  or \(\omega - \lambda \)  vanishes. We may assume that it is not true that both vanish, because this is covered by the case \(\omega , \lambda \in \mathbb {R}\) . Then \([a:b]=[0:1]\)  or \([a:b]=[1:0]\) .
1-dimensional Dirac operators
General formalism
By a 1d Dirac operator on the halfline we will mean a differential
operator of the form
\(D =\begin{bmatrix}a(x) & - \partial _x \\\partial _x & b(x)\end{bmatrix},\) 
where \(a,b\)  are smooth functions on \(\mathbb {R}_{+} = ]0,\infty [\) .
In this subsection we treat it as a formal operator acting, say, on the space of distributions on \(\mathbb {R}_{+} \)  valued
in 2.
We first describe a few integral kernels closely related to \(D\) .
Let \(k\in  and\begin{equation}\xi (k,x)=\begin{bmatrix}\xi _{\uparrow }(k,x)\\\xi _{\downarrow }(k,x)\end{bmatrix},\qquad \zeta (k,x)=\begin{bmatrix}\zeta _{\uparrow }(k,x)\\\zeta _{\downarrow }(k,x)\end{bmatrix}\end{equation}be a pair of linearly independent solutions of the Dirac equation:\begin{equation}(D-k)\xi (\cdot ,k)=(D-k)\zeta (\cdot ,k)=0.\end{equation}Let\begin{equation}d(k,x):=\det \left[\xi (k,x),\zeta (k,x)\right]=\det \begin{bmatrix}\xi _{\uparrow }(k,x)&\zeta _{\uparrow }(k,x)\\\xi _{\downarrow }(k,x)&\zeta _{\downarrow }(k,x).\end{bmatrix}\end{equation}Then \) d(k,x)\( does not depend on \) x\(, so that one can write \) d(k)\( instead.We define{\begin{@align}{1}{-1}& G^\leftrightarrow (k;x,y):=d(k)^{-1}\xi (k,x)\zeta (k,y)^ d(k)^{-1}\zeta (k,x)\xi (k,y)^[2ex]&=d(k)^{-1}\begin{bmatrix}\xi _{\uparrow }(k,x)\zeta _{\uparrow }(k,y)&\xi _{\uparrow }(k,x)\zeta _{\downarrow }(k,y)\\[1ex]\xi _{\downarrow }(k,x)\zeta _{\uparrow }(k,y)&\xi _{\downarrow }(k,x)\zeta _{\downarrow }(k,y) \end{bmatrix}- d(k)^{-1}\begin{bmatrix}\zeta _{\uparrow }(k,x) \xi _{\uparrow }(k,y)&\zeta _{\uparrow }(k,x)\xi _{\downarrow }(k,y)\\[1ex]\zeta _{\downarrow }(k,x)\xi _{\uparrow }(k,y)&\zeta _{\downarrow }(k,x)\xi _{\downarrow }(k,y)\end{bmatrix}. \end{@align}}Note that \) G(k,x,y)\( is uniquely defined by\begin{equation}(D-k) G^\leftrightarrow (k;x,y)=0,\qquad G^\leftrightarrow (k;x,x)=\begin{bmatrix}0&1\\-1&0 \end{bmatrix}.\end{equation}We will call it the {\em canonical bisolution.}\) We also have the forward and backward Green's operators given by the kernels
\(G^\rightarrow (k;x,y)=& G^\leftrightarrow (k;x,y){\mathchoice{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.5pt}l}{\rm 1\hspace{-2.77771pt}l}}_{\mathbb {R}_+}(x-y),\\G^\leftarrow (k;x,y)=& -G^\leftrightarrow (k;x,y){\mathchoice{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.5pt}l}{\rm 1\hspace{-2.77771pt}l}}_{\mathbb {R}_+}(y-x).\) 


They are uniquely defined by
\( (D-k) G^\rightarrow (k;x,y)=\delta (x-y) \begin{bmatrix}1&0\\0&1 \end{bmatrix}, &\quad x<y\Rightarrow G^\rightarrow (x,y)=0;\\(D-k) G^\leftarrow (k;x,y)=\delta (x-y) \begin{bmatrix}1&0\\0&1 \end{bmatrix},& \quad x>y\Rightarrow G^\leftarrow (x,y)=0.\) 


Note that \(G^\leftrightarrow , G^\leftarrow , G^\rightarrow \)  do not depend on the choice of \(\xi ,\zeta \) .
Using the eigensolutions \(\xi ,\zeta \) , we can introduce yet another important integral kernel:
\(G^{\bowtie }(k;x,y):=&-d(k)^{-1}\begin{bmatrix}\xi _{\uparrow }(k,x)\zeta _{\uparrow }(k,y)&\xi _{\uparrow }(k,x)\zeta _{\downarrow }(k,y)\\\xi _{\downarrow }(k,x)\zeta _{\uparrow }(k,y)&\xi _{\downarrow }(k,x)\zeta _{\downarrow }(k,y) \end{bmatrix}{\mathchoice{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.5pt}l}{\rm 1\hspace{-2.77771pt}l}}_{\mathbb {R}_+}(y-x)\\ & - d(k)^{-1}\begin{bmatrix}\zeta _{\uparrow }(k,x) \xi _{\uparrow }(k,y)&\zeta _{\uparrow }(k,x)\xi _{\downarrow }(k,y)\\\zeta _{\downarrow }(k,x)\xi _{\uparrow }(k,y)&\zeta _{\downarrow }(k,x)\xi _{\downarrow }(k,y)\end{bmatrix}{\mathchoice{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.22214pt}l}{\rm 1\hspace{-2.5pt}l}{\rm 1\hspace{-2.77771pt}l}}_{\mathbb {R}_+}(x-y).\) 

It is also Green's kernel, because
it satisfies
\((D-k) G^{\bowtie }(k;x,y)=\delta (x-y) \begin{bmatrix}1&0\\0&1 \end{bmatrix}.\) 

\( G^{\bowtie }(k;x,y)\) 
depends on the choice of the pair of 1-dimensional subspaces \((k, \cdot )\) ,
\((k, \cdot )\)  of \(\mathrm {Ker}(D-k)\) . The resolvents of various closed
realizations
of \(D\)  are often of this form.
Two classes of 1d Dirac operators have special properties.
The case \(a(x)=b(x)\)  can be fully diagonalized:
\( \begin{bmatrix}a(x) & - \partial _x \\\partial _x & a(x)\end{bmatrix}&=\frac{1}{\sqrt{2}}\begin{bmatrix}1&\mathrm {i}\\\mathrm {i}&1\end{bmatrix}\begin{bmatrix}-\mathrm {i}\partial _x+a(x)&0\\0&\mathrm {i}\partial _x+a(x)\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm {i}\\-\mathrm {i}&1\end{bmatrix}.\) 

We will analyze 1d Dirac-Coulomb operators of this form in Subsection
REF .
The case \(a(x)=-b(x)\)  can be brought to an antidiagonal
form, used in supersymmetry:
\(\begin{bmatrix}a(x) & - \partial _x \\\partial _x & -a(x)\end{bmatrix}&=\frac{1}{\sqrt{2}}\begin{bmatrix}1&-1\\1&1\end{bmatrix}\begin{bmatrix}0&-\partial _x-a(x)\\\partial _x-a(x)&0\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&1\\-1&1\end{bmatrix}.\) 

We will analyze 1d Dirac-Coulomb operators of this form in Subsection
.

Homogeneous first order scalar operators
Let \(\alpha \in . In this subsection we discuss the differential operator\begin{equation}A_{\alpha } = x^{\alpha } \partial _x x^{- \alpha } = \partial _x -\frac{ \alpha }{x}\end{equation}acting on scalar functions. It will be a building block of somespecial 1d Dirac-Coulomb operators considered in subsections\ref {app:elementary}and \ref {app.super}.\) Let us briefly recall basic results about realizations of
\(A_\alpha \)  as a closed operator in \(L^2(\mathbb {R}_+)\)  following
[1]}. Proofs of all statements stated in this subsection
without justification can be found therein.
(In [1]} a different convention was used:
\(A_{\alpha } = -\mathrm {i}\partial _x +\frac{\mathrm {i}\alpha }{x}\) . Thus
\(A_\alpha ^{\rm new}=\mathrm {i}A_\alpha ^{\rm old}\) .)
We let \(A_{\alpha }^{\min }\)  be the closure (in the sense of operators on \(L^2(\mathbb {R}_+)\) ) of the restriction of \(A_{\alpha }\)  to \(C_c^{\infty }(\mathbb {R}_+)\)  and \(A_{\alpha }^{\max }\)  the restriction of \(A_{\alpha }\)  to \(\mathrm {Dom}(A_{\alpha }^{\max }) = \lbrace  f \in L^2(\mathbb {R}_+) \, | \, A_{\alpha } f \in L^2(\mathbb {R}) \rbrace \) . Operators \(A_{\alpha }^{\min }\)  and \(-A_{ -\overline{\alpha }}^{\max }\)  are adjoint to each other.
Proposition 40 
We have \(A_{\alpha }^{\min } = A_{\alpha }^{\max }\)  if and only if
\(|\mathrm {Re}(\alpha )| \ge \frac{1}{2}\) . If \(|\mathrm {Re}(\alpha )| < \frac{1}{2}\) , then
\(\mathrm {Dom}(A_{\alpha }^{\max }) = \mathrm {Dom}(A_{\alpha }^{\min }) + x^{\alpha }\) , where \(\chi \in C_\mathrm {c}^\infty (\mathbb {R}_+)\)  and \(\chi =1\) 
near 0.
If \(\mathrm {Re}(\alpha ) \ne \frac{1}{2}\) , then \(\mathrm {Dom}(A_{\alpha }^{\min })= H_0^1(\mathbb {R}_+)\) .

Closed realizations of \(A_\alpha \)  are of two types,
described in the following pair of propositions.
Proposition 41 Let \(\mathrm {Re}(\alpha ) > - \frac{1}{2}\) ,

\(\sigma (-\mathrm {i}A_{\alpha }^{\max }) = \mathbb {R}\cup +\) 
and one has
\((A^{\max }_{\alpha } - \mathrm {i}k)^{-1}f (x) = - \int _{x}^{\infty } \mathrm {e}^{\mathrm {i}k(x-y)} \left( \frac{x}{y} \right)^{\alpha } f(y) y̥, \qquad \mathrm {Im}(k) < 0 .\) 

If \(k \in +\)  and \(n \ge 1\) , then \(\mathrm {Ker}\left( (A_{\alpha }^{\max } - \mathrm {i}k)^n \right)\)  is the space of functions of
the form \(x^{\alpha } \mathrm {e}^{\mathrm {i}k x} q(x)\)  with \(q\)  polynomial of degree
at most \(n-1\) . In particular \(\bigcup \limits _{n=0}^{\infty }\mathrm {Ker}\left( ( A_{\alpha }^{\max } - \mathrm {i}k)^n \right)\)  is dense in
\(L^2(\mathbb {R}_+)\) .

If \(k \in +\) , then \(A_{\alpha }^{\max } - \mathrm {i}k\)  is a Fredholm operator of index 1.


Proposition 42 Let \(\mathrm {Re}(\alpha ) < \frac{1}{2}\) .

\(\sigma (-\mathrm {i}A_{\alpha }^{\min }) = \mathbb {R}\cup -\)  and one has
\((A^{\min }_{\alpha } - \mathrm {i}k)^{-1}f (x) = \int _{0}^{x} \mathrm {e}^{\mathrm {i}k(x-y)} \left( \frac{x}{y} \right)^{\alpha } f(y) y̥, \qquad \mathrm {Im}(k)> 0 .\) 

\(A_{\alpha }^{\min }\)  has no eigenvectors.

If \(k \in -\) , then \(A_{\alpha }^{\min } - \mathrm {i}k\)  is a Fredholm
operator of index \(-1\) .


Proof of Propositions REF  and REF .
Statements 1. are proven in [1]}.
2. requires justification only for the first part in the first proposition. We factorize
\(x^{\alpha } \mathrm {e}^{\mathrm {i}k x} q(x) = x^{\mathrm {i}\, \mathrm {Im}(\alpha )} \mathrm {e}^{\mathrm {i}\, \mathrm {Re}(k) x} \left( x^{\mathrm {Re}(\alpha )} \mathrm {e}^{- \mathrm {Im}(k) x} q(x) \right).\) 
Functions in the parenthesis form a dense set, because for any real
numbers \(c >0\) , \(\beta > - 1\)  functions \(e^{- \frac{c x}{2} }x^{\frac{\beta }{2}} L_n^{(\beta )}(cx)\) , with \(L_n^{(\beta )}\) 
Laguerre polynomials, form an orthogonal basis (see e.g. [4]}). Clearly density is unaffected by the prefactor, which amounts to the action of a certain unitary operator on \(L^2\) .
Let us show 3. We consider first the case \(|\mathrm {Re}(\alpha )| < \frac{1}{2}\) . Then we have explicit inverses modulo rank one operators.
If \(\mathrm {Im}(k)>0\) , then \(A^{\min }_{\alpha } - \mathrm {i}k\)  is invertible and its inverse is a right inverse for \(A_{\alpha }^{\max } - \mathrm {i}k\) . Thus \(A_{\alpha }^{\max } - \mathrm {i}k\)  is surjective. We already know that its kernel is one-dimensional.
If \(\mathrm {Im}(k)<0\) , then \((A_{\alpha }^{\max } - \mathrm {i}k)^{-1} : L^2(\mathbb {R}_+) \rightarrow \mathrm {Dom}(A_{\alpha }^{\max })\)  is continuous. The range of \(A_{\alpha }^{\min } - \mathrm {i}k\)  is the preimage of \(\mathrm {Dom}(A_{\alpha }^{\min })\) , which is a closed subspace of \(\mathrm {Dom}(A_{\alpha }^{\max })\)  of codimension one. Hence \(\mathrm {Ran}(A_{\alpha }^{\min } - \mathrm {i}k)\)  is a closed subspace of \(L^2(\mathbb {R}_+)\)  of codimension one.
In the region \(|\mathrm {Re}(\alpha )| \ge \frac{1}{2}\)  we have \(A_{\alpha }^{\min } = A_{\alpha }^{\max }\) , so the above argument does not generalize. To extend the result, note that integral kernels of \((A_{\alpha }^{\min } - \mathrm {i}k)^{-1} -(A_{\beta }^{\min } - \mathrm {i}k)^{-1}\)  (resp. \((A_{\alpha }^{\max } - \mathrm {i}k)^{-1} - (A_{\beta }^{\max } - \mathrm {i}k)^{-1}\) ) are square integrable for
\(\mathrm {Re}(\alpha ), \, \mathrm {Re}(\beta ) < \frac{1}{2}\)  and \(k \in +\) 
(resp. \(\mathrm {Re}(\alpha ), \, \mathrm {Re}(\beta ) > - \frac{1}{2}\)  and \(k \in -\) ). Hence by Lemma REF  we have that
\(A_{\alpha }^{\max } - \mathrm {i}k\)  and \(A_{\alpha }^{\min } - \mathrm {i}k\)  is Fredholm
whenever \(\mathrm {Im}(k) \ne 0\) . The statement about the value of the index is clear.
\(\Box \)
Proposition 43 
\( A_{\alpha }^{\max }\)  is the generator of a \(C^0\) -semigroup if and only if \(\mathrm {Re}(\alpha ) \ge 0\) . If this condition is satisfied, it generates the semigroup of contractions
\((\mathrm {e}^{ t A_{\alpha }^{\max }} f)(x) = x^{\alpha } (x+t)^{- \alpha } f(x+t), \qquad t \ge 0,\) 
\(- A_{\alpha }^{\min }\)  is the generator of a \(C^0\) -semigroup if and only if \(\mathrm {Re}(\alpha ) \le 0\) . If this condition is satisfied, it generated a semigroup of contractions
\((\mathrm {e}^{- t A_{\alpha }^{\min }}f)(x) = x^{\alpha } (x-t)^{- \alpha } f(x-t), \qquad t \ge 0.\) 
Here we put \(f(x-t)=0\)  if \(x-t<0\) .
If \(|\mathrm {Re}(\alpha )| < \frac{1}{2}\) , the operators \(- A_{\alpha }^{\max }\)  and \(A_{\alpha }^{\min }\)  are not generators of \(C^0\) -semigroups.

We present a proof of the statements concerning
\(A_{\alpha }^{\max }\) . The others can be proven analogously. It is
elementary to check that for \(\mathrm {Re}(\alpha ) \ge 0\)  the right hand side
of (REF ) defines a \(C^0\) -semigroup of contractions
with the generator \(A_{\max }^{\alpha }\) . If \(\mathrm {Re}(\alpha ) < 0\) , we consider the same expression for \(f \in C_c^{\infty }(]1, \infty [)\) . Then for \(t \le 1\)  it is the unique solution of the Cauchy problem \( \frac{{t̥} f_t = A_{\alpha }^{\max } f_t, f_0 = f. However, there exists no constant c such that \Vert  f_t \Vert  \le c \Vert  f \Vert  for every t \in [0,1] and f. Thus  A_{\alpha }^{\max } is not a generator. If~|\mathrm {Re}(\alpha )| < \frac{1}{2}, then \sigma ( - A_{\alpha }^{\max }) is the right closed complex half-plane, so - A_{\alpha }^{\max } is not a generator.}{}\)

Dirac-Coulomb Hamiltonians with \(\omega =0\)
Dirac-Coulomb Hamiltonians with \(\omega =0\)  can be reduced to operators
\(A_\alpha \)  studied in Subsection REF .
Therefore, they can be analyzed using elementary functions only.
Let us set \(W:=\frac{1}{\sqrt{2}} \begin{bmatrix} 1 & \mathrm {i}\\ \mathrm {i}& 1 \end{bmatrix} \) .
Using (REF ) we obtain for all \(\lambda \)
\(D_{0,\lambda }^{\min } = W \begin{bmatrix}-\mathrm {i}A_{\mathrm {i}\lambda }^{\min } & 0\\ 0 & \mathrm {i}A_{- \mathrm {i}\lambda }^{\min } \end{bmatrix} W^{-1},\qquad D_{0,\lambda }^{\max } = W \begin{bmatrix}-\mathrm {i}A_{\mathrm {i}\lambda }^{\max } & 0\\ 0 & \mathrm {i}A_{- \mathrm {i}\lambda }^{\max } \end{bmatrix} W^{-1}.\) 
Consider now the homogeneous holomorphic family. Note first that
\(\omega =0\)  implies \(\mu =\pm \mathrm {i}\lambda \) .
We set \(D_{\lambda }^{\pm }:=D_{0,\lambda , \pm \mathrm {i}\lambda , [\mp \mathrm {i}: 1]}\) . Note that \((0,\lambda , \pm \mathrm {i}\lambda , [\mp \mathrm {i}: 1])\in \mathcal {E}^{\pm }\) . We have:
\(D_{\lambda }^+ &= W \begin{bmatrix}-\mathrm {i}A_{\mathrm {i}\lambda }^{\max } & 0 \\ 0 & \mathrm {i}A_{- \mathrm {i}\lambda }^{\min } \end{bmatrix} W^{-1}, \qquad \mathrm {Re}(\mathrm {i}\lambda ) > - \frac{1}{2}, \\D_{\lambda }^- &= W \begin{bmatrix} -\mathrm {i}A_{\mathrm {i}\lambda }^{\min } & 0 \\ 0 & \mathrm {i}A_{- \mathrm {i}\lambda }^{\max } \end{bmatrix} W^{-1}, \qquad \mathrm {Re}(- \mathrm {i}\lambda ) > - \frac{1}{2},\) 
Below \(\sigma _2\)  is the Pauli matrix \(\begin{bmatrix} 0 & -\mathrm {i}\\ \mathrm {i}& 0 \end{bmatrix}\) . Matrices \(\frac{1 \pm \sigma _2}{2}\)  are its spectral projections.
Proposition 44 We have \(\sigma (D_{\lambda }^+) = \mathbb {R}\cup +\)  and
\((D_{\lambda }^+ - k)^{-1} = (-\mathrm {i}A_{\mathrm {i}\lambda }^{\max }-k )^{-1}\frac{1+\sigma _2}{2} - (-\mathrm {i}A_{- \mathrm {i}\lambda }^{\min }+k)^{-1} \frac{1 - \sigma _2}{2}, \qquad \mathrm {Im}(k)<0,\) 
whereas \(\sigma (D_{\lambda }^-) = \mathbb {R}\cup -\)  and
\((D_{\lambda }^- - k)^{-1} = (-\mathrm {i}A_{\mathrm {i}\lambda }^{\min }-k )^{-1}\frac{1+ \sigma _2}{2} - (-\mathrm {i}A_{- \mathrm {i}\lambda }^{\max }+k)^{-1} \frac{1 - \sigma _2}{2}, \qquad \mathrm {Im}(k)>0.\) 

Follows from identities (REF ) and Proposition REF , REF .
Proposition 45 \(D_{\lambda }^{\pm } - k \)  with \(k \in {\pm }\)  are Fredholm of
index 0.

Indeed, by Propositions REF  and REF 
they are direct sums of two Fredholm operators with indices 1 and
\(-1\) .
Proposition 46 Let \(k \in \pm \) . Then \(\bigcup \limits _{n=0}^{\infty } \mathrm {Ker}( (D_{\lambda }^\pm - k)^n)\)  is a dense subspace of \(L^2(\mathbb {R}_+) \begin{bmatrix} \mp \mathrm {i}\\ 1 \end{bmatrix}\) .

Proposition 47 \( \mathrm {i}D_{\lambda }^+\)  is the generator of a \(C^0\) -semigroup if and only if \(\mathrm {Im}(\lambda ) \le 0\) . Then it generates the semigroup of contractions
\(\mathrm {e}^{\mathrm {i}t D_{\lambda }^+} = \mathrm {e}^{ t A_{\mathrm {i}\lambda }^{\max }} \frac{1 + \sigma _2}{2} + \mathrm {e}^{- t A^{\min }_{- \mathrm {i}\lambda }} \frac{1 - \sigma _2}{2}, \qquad t \ge 0.\) 
\(- \mathrm {i}D_{\lambda }^-\)  is the generator of a \(C^0\) -semigroup if and only if \(\mathrm {Im}(\lambda ) \ge 0\) . Then it generates the semigroup of contractions
\(\mathrm {e}^{-\mathrm {i}t D_{\lambda }^-} = \mathrm {e}^{- t A_{\mathrm {i}\lambda }^{\min }} \frac{1 + \sigma _2}{2} + \mathrm {e}^{ t A^{\max }_{- \mathrm {i}\lambda }}\frac{1 - \sigma _2}{2}, \qquad t \ge 0.\) 
Operators \(- \mathrm {i}D_{\lambda }^+\)  and \(\mathrm {i}D_{\lambda }^-\)  are not generators of \(C^0\) -semigroups.


Hankel transformation
The following proposition is proven e.g. in [1]}.
Proposition 48 Let \(\mathrm {Re}(m) \ge -1\) . We define
\((\mathcal {F}_m^{\mathrm {pre}} f)(x) = \int _0^{\infty } J_m(xy) \sqrt{xy} f(y) y̥, \qquad f \in C_c^{\infty }(\mathbb {R}_+),\) 
where \(J_m\)  is the Bessel function. Then \(\mathcal {F}_m^{\mathrm {pre}}\)  extends
to a bounded operator \(\mathcal {F}_m\)  on \(L^2(\mathbb {R}_+)\) , known as the Hankel transformation. \(\mathcal {F}_m\)  is a self-transposed involution, unitary if \(m\)  is real.

Recall that the operator \(X\)  is defined by
\( (Xf)(x)= x f(x), \qquad \mathrm {Dom}(X) = \lbrace  f \in L^2(\mathbb {R}_+) \, | \, x f(x) \in L^2(\mathbb {R}_+) \rbrace . \) 
Proposition 49 If \(\mathrm {Re}(\alpha ) > - \frac{1}{2}\) , one has
\(\mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}} = -X, \qquad \mathcal {F}_{\alpha - \frac{1}{2}} A_{- \alpha }^{\min } \mathcal {F}_{\alpha + \frac{1}{2}} = X.\) 

Using the identity
\(x^{-m} \frac{{x̥} x^m J_m(x) =J_{m-1}(x)}{o}ne checks that\begin{equation}(\mathcal {F}_{\alpha + \frac{1}{2}}^{\mathrm {pre}} A_{\alpha } f)(x) = - x (\mathcal {F}_{\alpha - \frac{1}{2}}^{\mathrm {pre}} f)(x)\end{equation}for f \in C_c^{\infty }(\mathbb {R}_+). If |\mathrm {Re}(\alpha )| < \frac{1}{2}, (\ref {eq:AF_intertwine}) may be checked to hold also for f(x) = \chi (x) x^{\alpha }. Taking closures we obtain \mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \subset - X \mathcal {F}_{\alpha - \frac{1}{2}}, so \mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}} \subset - X. Since \mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}} is a~closed operator and C_c^{\infty }(\mathbb {R}_+) is a dense subspace of \mathrm {Dom}(X) with respect to the graph topology, the opposite inclusion will be established by demonstrating that C_c^{\infty }(\mathbb {R}_+) \subset \mathrm {Dom}(\mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}}). Let f \in C_c^{\infty }(\mathbb {R}_+). It is clear that \mathcal {F}_{\alpha - \frac{1}{2}} f is a smooth function. Using the identity \left( x \frac{{x̥} - y \frac{{y̥}  \sqrt{xy} J_m(xy) =0 we find\begin{equation}\frac{{x̥} (\mathcal {F}_{\alpha - \frac{1}{2}} f)(x) = \frac{1}{x} \int _0^{\infty } \sqrt{xy} J_m(xy) \frac{{y̥} \frac{f(y)}{y} y̥.}{S}ince \frac{{y̥} \frac{f(y)}{y} is in L^2(\mathbb {R}_+), we get that \frac{{x̥} (\mathcal {F}_{\alpha - \frac{1}{2}} f)(x) is square-integrable over [1 , \infty [. Next we use the series expansion of J_m to find that for small x\begin{equation}(\mathcal {F}_{\alpha - \frac{1}{2}} f)(x) = \frac{\int _0^{\infty } y^{\alpha } f(y) y̥}{2^{\alpha - \frac{1}{2}} \Gamma (\alpha + \frac{1}{2})} x^{\alpha } + O(x^{\alpha +1}).\end{equation}Hence \mathcal {F}_{\alpha - \frac{1}{2}} f \in \mathrm {Dom}(A_{\alpha }^{\max }), so f \in \mathrm {Dom}(A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}}) = \mathrm {Dom}(\mathcal {F}_{\alpha + \frac{1}{2}} A_{\alpha }^{\max } \mathcal {F}_{\alpha - \frac{1}{2}}). We proved the first equality in (\ref {eq:AF_id}). The other one may be obtained by taking the transpose.}{}}{F}ollowing \cite {DeRi17_01} (see also \cite {BuDeGe11_01}), we consider the formal differential operator\begin{equation}L_{m^2} = - \partial _x^2 + \frac{m^2 - \frac{1}{4}}{x^2}.\end{equation}We let L_{m^2}^{\min } be the closure of its restriction toC_c^{\infty }(\mathbb {R}_+) and L_{m^2}^{\max } be the restriction to\mathrm {Dom}(L_{m^2}^{\max }) = \lbrace  f \in L^2(\mathbb {R}_+) \, | \, L_{m^2} f \in L^2(\mathbb {R}_+)\rbrace . If \mathrm {Re}(m) > -1, operator H_m is defined as the restriction ofL_{m^2} to \mathrm {Dom}(L_{m^2}^{\min }) + x^{m+ \frac{1}{2}},where \chi  is a smooth function equal to one in a~neighborhood ofzero. We remark that H_{\frac{1}{2}} and H_{-\frac{1}{2}} are theDirichlet Laplacian and the Neuman Laplacian,respectively. Furthermore, H_m can be diagonalized as follows:\begin{equation}H_m = \mathcal {F}_m X^2 \mathcal {F}_m.\end{equation}}{}\end{equation}}{\subsection }{Dirac-Coulomb Hamiltonians with \lambda =0}}{D}irac-Coulomb Hamiltonians with \right.\lambda =0can be analyzed without Whittaker functions, just with Bessel functions.\) 
Let us set \(U := \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & -1 \\ 1 & 1 \end{bmatrix}\) .
Using (REF ) we obtain for all \(\omega \)
\(D_{\omega ,0}^{\min } = U \begin{bmatrix} 0 & - A_{\omega }^{\min } \\ A_{- \omega }^{\min } & 0 \end{bmatrix} U^{-1},\qquad D_{\omega ,0}^{\max } = U \begin{bmatrix} 0 & - A_{\omega }^{\max } \\ A_{- \omega }^{\max } & 0 \end{bmatrix} U^{-1}.\) 
Using Proposition REF  we rewrite the operators \(D^{\prime \pm }_{\omega } :=D_{\omega , 0 , \pm \omega , [ \mp 1 : 1 ]}\) 
as
\(D^{\prime +}_{\omega } &= U \begin{bmatrix} 0 & - A_{\omega }^{\max } \\A_{- \omega }^{\min } & 0 \end{bmatrix} U^{-1},\quad \mathrm {Re}(\omega )>-\frac{1}{2}; \\D^{\prime -}_{\omega } &= U \begin{bmatrix} 0 & - A_{\omega }^{\min } \\ A_{- \omega }^{\max } & 0 \end{bmatrix} U^{-1},\quad -\mathrm {Re}(\omega )>-\frac{1}{2}.\) 
Proposition 50 Introduce
\(\mathcal {W}_\omega ^{\prime \pm }:=\frac{1}{2}\begin{bmatrix} \mathcal {F}_{\pm \omega \pm \frac{1}{2}}{+}\mathcal {F}_{\pm \omega \mp \frac{1}{2}}& \mathcal {F}_{\pm \omega \pm \frac{1}{2}}{-}\mathcal {F}_{\pm \omega \mp \frac{1}{2}} \\\mathcal {F}_{\pm \omega \pm \frac{1}{2}}{-}\mathcal {F}_{\pm \omega \mp \frac{1}{2}} &\mathcal {F}_{\pm \omega \pm \frac{1}{2}}{+}\mathcal {F}_{\pm \omega \mp \frac{1}{2}} \end{bmatrix},\quad \pm \mathrm {Re}(\omega )>-\frac{1}{2}.\) 
Then
\( \mathcal {W}_\omega ^{\prime \pm }\)  are involutions and
we have the following diagonalizations
\(D_{\omega }^{\prime \pm }=\mathcal {W}_\omega ^{\prime \pm }\begin{bmatrix}\mp X & 0 \\ 0&\pm X \end{bmatrix}\mathcal {W}_\omega ^{\prime \pm }.\) 

We insert (REF ) into (REF ):
\(D_{\omega }^{\prime +} = U \begin{bmatrix} \mathcal {F}_{\omega + \frac{1}{2}} & 0 \\ 0 & \mathcal {F}_{\omega - \frac{1}{2}} \end{bmatrix} \begin{bmatrix} 0 & X \\ X & 0 \end{bmatrix} \begin{bmatrix} \mathcal {F}_{\omega + \frac{1}{2}} & 0 \\ 0 & \mathcal {F}_{\omega - \frac{1}{2}} \end{bmatrix} U^{-1}, \\D_{\omega }^{\prime -} = U \begin{bmatrix} \mathcal {F}_{-\omega - \frac{1}{2}} & 0 \\ 0 & \mathcal {F}_{-\omega + \frac{1}{2}} \end{bmatrix} \begin{bmatrix} 0 & -X \\ -X & 0 \end{bmatrix} \begin{bmatrix} \mathcal {F}_{-\omega - \frac{1}{2}} & 0 \\ 0 & \mathcal {F}_{-\omega + \frac{1}{2}} \end{bmatrix} U^{-1}.\) 

Then we use
\(\begin{bmatrix}0&\pm X\\\pm X&0\end{bmatrix}=U^{-1}\begin{bmatrix}\mp X&0\\0&\pm X\end{bmatrix}U.\) 
Corollary 51 We have
\((D^{\prime +}_{\omega })^2 = U \begin{bmatrix} H_{\omega + \frac{1}{2}} & 0 \\ 0 & H_{\omega - \frac{1}{2}} \end{bmatrix} U^{-1}, \\(D^{\prime -}_{\omega })^2 = U \begin{bmatrix} H_{- \omega - \frac{1}{2}} & 0 \\ 0 & H_{- \omega + \frac{1}{2}} \end{bmatrix} U^{-1}.\) 

Remark 52 At least formally, operators \(D_{\omega }^{\prime \pm }\) , \(X \sigma _2 \) 
(declared to be odd) and \((D_{\omega }^{\prime \pm })^2\) , \(X^2\) , \(A\)  (declared
to be even) furnish a representation of the Lie superalgebra
\(\mathfrak {osp}(1|2)\) . We leave a detailed description of this
representation for a future study.


Dirac Hamiltonian in \(d\)  dimensions
Separation of variables of a spherically symmetric Dirac Hamiltonian
in dimension 3
is described in many texts and belongs to the standard curriculum of
relativistic quantum mechanics [6]}.
Of course, it is even more
straightforward to solve a rotationally symmetric Dirac Hamiltonian in
dimension 2. However, to our knowledge, the first treatment
in any dimension is due to Gu, Ma and Dong [7]}.
In this appendix we show that a spherically symmetric Dirac Hamiltonian
in an arbitrary dimension can be reduced to 1 dimension.
Unlike in [7]},
we arrive at the radial Dirac equation by relatively simple
algebraic computations which do not involve a detailed analysis of
representations of the Lie algebra
\(\mathfrak {so}(d)\) .
The main role in this separation is played by a certain operator
\(\kappa \)  that commutes with the Dirac operator. This operator in dimension 3
goes back to Dirac himself.
It seems that for the
first time it has been generalized to other dimensions in
[7]}. We analyze this operator in detail.
Recall that operators belonging to the center of the envelopping
algebra of
\(\mathfrak {so}(d)\)  are called Casimir operators of \(\mathfrak {so}(d)\) .
One of them, built in a standard way as a bilinear form in the
generators, will be called the square of angular momentum or simpy the quadratic Casimir (even though it is not the only Casimir bilinear in generators: these form a vector space generically of dimension 1, and of dimension 2 if \(d=4\) ).
\(\kappa \)  does not coincide with the quadratic Casimir.
One can ask whether \(\kappa \)  is also a Casimir operator.
We will analyse this question in detail. It turns out that the answer
is positive in even, and negative in odd dimensions.
Laplacian in \(d\)  dimensions
Spherical coordinates can be interpreted as a map
\(\mathbb {R}^d\backslash \lbrace 0\rbrace \ni x&\mapsto (\hat{x},r)\in \mathbb {R}_+\times \mathbb {S}^{d-1},\\\hat{x}=\frac{x}{|x|},&\quad r=|x|.\) 


It induces a unitary map
\(L^2(\mathbb {R}^d)\rightarrow L^2(\mathbb {R}_+,r^{d-1})\otimes L^2( \mathbb {S}^{d-1}).\) 
We also have the obvious map
\(L^2(\mathbb {R}_+,r^{d-1})\ni f\mapsto r^{\frac{d-1}{2}}f\in L^2(\mathbb {R}_+).\) 
The product of (REF ) and (REF )
will be denoted
\(U:L^2(\mathbb {R}^d)\rightarrow L^2(\mathbb {R}_+)\otimes L^2( \mathbb {S}^{d-1}).\) 
The momentum is defined as
\(p_i:=-\mathrm {i}\partial _i.\) 
We also introduce the radial momentum
\(R &:=\frac{x}{2|x|}p+p\frac{x}{2|x|}=\frac{x}{|x|}p-\mathrm {i}\frac{(d-1)}{2|x|}.\) 

Here is the radial momentum and its square in spherical coordinates:
\(R& =-\mathrm {i}\partial _r-\mathrm {i}\frac{d-1}{2r},\\R ^2&=-\partial _r^2-\frac{d-1}{r}\partial _r+ \left(\frac{1}{4}-\Big (\frac{d-2}{2}\Big )^2\right)\frac{1}{r^2}.\) 


After applying \(U\)  we obtain
\(URU^{-1}=-\mathrm {i}\partial _r.\) 
In the standard way we introduce the angular momentum and its square:
\(L_{ij}&:=x_ip_j-x_jp_i,\\L^2&:=\sum _{i<j}L_{ij}^2.\) 


They furnish the standard representation of the Lie algebra \(\mathfrak {so}(d)\)  in the
on \(\mathbb {S}^{d-1}\) :
\([L_{ij},x_k]&=-\mathrm {i}\delta _{jk}x_i+\mathrm {i}\delta _{ik}x_j,\\[L_{ij},p_k]&=-\mathrm {i}\delta _{jk}p_i+\mathrm {i}\delta _{ik}p_j,\\\Big [L_{ij},L_{kl}\Big ]&=-\mathrm {i}\delta _{jk}L_{il}-\mathrm {i}\delta _{il}L_{jk}+\mathrm {i}\delta _{ik}L_{jl}+\mathrm {i}\delta _{jl}L_{ik}.\) 


The angular momentum squared \(L^2\)  is the quadratic Casimir operator of \(\mathfrak {so}(d)\) .
The representation (REF ) is decomposed into subspaces of
spherical harmonics of the order \(\ell \) . The
representation
of \(\mathfrak {so}(d)\)  of this type will be called spherical of degree
\(\ell \). On this representation we have
\(L^2=&\ell (\ell +d-2)=\Big (\ell +\frac{d-2}{2}\Big )^2-\Bigl (\frac{d-2}{2}\Bigr )^2.  \) 
The Laplacian on \(\mathbb {R}^d\)  in the spherical coordinates is
\(-\Delta &=-\partial _r^2-\frac{(d-1)}{r}\partial _r+\frac{L_d^2}{r^2},\\&=R ^2+\Big (-\frac{1}{4}+\Big (\ell +\frac{d-2}{2}\Big )^2\Big )\frac{1}{r^2}.\) 


Sandwiching with \(U\)  we obtain
\(U(-\Delta )U^{-1}=-\partial _r^2+\Big (-\frac{1}{4}+\Big (\ell +\frac{d-2}{2}\Big )^2\Big )\frac{1}{r^2}.\) 
Remark 53 Discussion above is valid even for \(d=1\) , with \(\mathbb {S}^0 := \lbrace  \pm 1 \rbrace \) . This case is peculiar in that the only allowed values of \(\ell \)  are 0 and 1, corresponding to even and odd functions. \(d=2\)  is also special: \(\ell \)  takes arbitrary integer values, while for \(d \ge 3\)  one has \(\ell \ge 0\) .


Dirac operator in \(d\)  dimensions
Let \(\alpha _i\) , \(i=1, \ldots ,d\)  and \(\beta \)  be the Clifford matrices
acting irreducibly in a finite dimensional space \(\mathcal {K}\) . They satisfy the Clifford relations
\([\alpha _i,\alpha _j]_+=2\delta _{ij},\quad [\alpha _i,\beta ]_+=0,\quad \beta ^2=1.\) 
We recall that \(\dim (\mathcal {K}) = 2^{\lfloor \frac{d+1}{2} \rfloor }\)  and that for even \(d\)  one has \(\beta = \pm \mathrm {i}^{\frac{d}{2}}\prod \limits _{i=1}^d \alpha _i\) . The two sign choices give non-isomorphic representations of the Clifford algebra.
Using the Einstein summation convention unless there is a summation
sign, we introduce the following operators on \(L^2(\mathbb {R}^d)\otimes \mathcal {K}\) :
\(D&:=\alpha _i p_i,\\T&:=-\mathrm {i}\sum _{i<j}\alpha _i\alpha _j L_{ij}+\frac{d-1}{2},  \\S&:=\frac{\alpha _ix_i}{|x|},\\\kappa &:=\beta T=T\beta .\) 
Proposition 54 We have
\(SD&=R+\frac{\mathrm {i}}{|x|}T,&DS&=R-\frac{\mathrm {i}}{|x|}T,\\SR&=RS,& ST&=-TS,\\S\beta &=-\beta S& \beta D&=-D \beta ,\\DT&=-TD,& D\kappa &=\kappa D.\) 

Let us prove the first identity of
(REF ). We have
\(& [\alpha _i\alpha _jL_{ij},\alpha _kp_k]_+\\=&[\alpha _i\alpha _j,\alpha _k]L_{ij}p_k+\alpha _k \alpha _i\alpha _j[L_{ij},p_k]_+. \nonumber \) 

Using
\(\alpha _j\alpha _k\alpha _i&=\alpha _k\alpha _i\alpha _j+2\delta _{jk}\alpha _i-2\delta _{ij}\alpha _k,\\\alpha _i\alpha _j\alpha _k&=\alpha _k\alpha _i\alpha _j+2\delta _{jk}\alpha _i-2\delta _{ik}\alpha _j, \) 


we obtain
\(&3\alpha _k\alpha _i\alpha _j[L_{ij},p_k]_+ \nonumber \\= &\alpha _k\alpha _i\alpha _j[L_{ij},p_k]_+ +\alpha _j\alpha _k\alpha _i[L_{ki},p_j]_++\alpha _i\alpha _j\alpha _k[L_{jk},p_i]_+ \nonumber \\=& \alpha _k\alpha _i\alpha _j\big ([L_{ij},p_k]_++[L_{ki},p_j]_++[L_{jk},p_i]_+\big ) \nonumber \\+ & 2\alpha _i[L_{ji},p_j]_+-2\alpha _i[L_{ij},p_j]_+-2\alpha _i[L_{ij},p_i]_+ \nonumber \\=&6\alpha _i[L_{ji},p_j]_+=-6 \mathrm {i}\alpha _jp_j (d-1)-12\alpha _i L_{ij} p_j.\) 

Moreover,
\([\alpha _i\alpha _j,\alpha _k]L_{ij}p_k&=2(\alpha _i\delta _{jk}-\delta _{ik}\alpha _j)L_{ij}p_k \nonumber \\&=4\alpha _iL_{ij}p_j.\) 

Now the sum of \(\frac{\mathrm {i}}{6}\) (REF ) and \(\frac{\mathrm {i}}{2}\) (REF ) is \((d-1)\alpha _ip_i\) .
\(\Box \)

Decomposition into incoming and outgoing Dirac waves
Let
\(\Pi _\pm :=\frac{1}{2}(1\pm S)\) 
be the spectral projections of \(S\)  onto \(\pm 1\) . Define
\(\mathcal {H}_\pm :=\Pi _{\pm }(L^2(\mathbb {R}^d)\otimes \mathcal {K}),\ \text{so that }\ L^2(\mathbb {R}^d)\otimes \mathcal {K}=\mathcal {H}_+\oplus \mathcal {H}_-.\) 
For an operator \(B\)  on \(L^2(\mathbb {R}^d)\otimes \mathcal {K}\)  let us write
\(B_{\pm \pm }=\Pi _\pm B\Pi _\pm ,\qquad B_{\pm \mp }=\Pi _\pm B\Pi _\mp .\) 
Clearly,
\(S=\begin{bmatrix}1&0\\0&-1\end{bmatrix},&\qquad R=\begin{bmatrix}R_{++}&0\\0&R_{--}\end{bmatrix},\\\beta =\begin{bmatrix}0&\beta _{+-}\\\beta _{-+}&0\end{bmatrix},&\qquad T=\begin{bmatrix}0&T_{+-}\\T_{-+}&0\end{bmatrix},\\D =\begin{bmatrix}R_{++}&\frac{\mathrm {i}}{|x|}T_{+-}\\-\frac{\mathrm {i}}{|x|}T_{-+}&-R_{--}\end{bmatrix},&\qquad \kappa =\begin{bmatrix}\beta _{+-}T_{-+}&0\\0&\beta _{-+}T_{+-}\end{bmatrix}= \begin{bmatrix}T_{+-}\beta _{-+}&0\\0&T_{-+}\beta _{+-}\end{bmatrix}.\) 


\(D\)  commutes with the self-adjoint operator \(\kappa \) . We can
therefore reduce ourselves to the eigenspace of \(\kappa \)  with eigenvalue
\(\omega \in \mathbb {R}\) , denoted \(\mathcal {H}_\omega \) .
We can write
\(D =\begin{bmatrix}R_{++}&\frac{\mathrm {i}\omega }{|x|}\beta _{+-}\\-\frac{\mathrm {i}\omega }{|x|}\beta _{-+}&-R_{--}\end{bmatrix}.\) 
Using spherical coordinates, we can identify
\(L^2(\mathbb {R}^d)\otimes \mathcal {K}\)  with
\( L^2(\mathbb {R}_+,r^{d-1})\otimes L^2(\mathbb {S}^{d-1})\otimes \mathcal {K}\) . Applying (REF ) and treating \(\beta _{\pm \mp }\)  as
identifications, we can rewrite the above equation as
\(D =\begin{bmatrix}-\mathrm {i}\partial _r&\frac{\mathrm {i}\omega }{r}\\-\frac{\mathrm {i}\omega }{r}&\mathrm {i}\partial _r\end{bmatrix}.\) 
Consider now the Dirac Hamiltonian with a spherically symmetric potential
\(V(r)\)  and mass \(m(r)\) .
Reduction (REF ) leads to
\( D+V(r)+m(r)\beta =\begin{bmatrix}-\mathrm {i}\partial _r+V(r)&\frac{\mathrm {i}\omega }{r}+m(r)\\-\frac{\mathrm {i}\omega }{r}+m(r)&\mathrm {i}\partial _r +V(r)\end{bmatrix}.\) 
We prefer another form, related by a similarity transformation:
\( \frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm {i}\\\mathrm {i}&-1\end{bmatrix}\begin{bmatrix}-\mathrm {i}\partial _r+V(r)&\frac{\mathrm {i}\omega }{r}+m(r)\\-\frac{\mathrm {i}\omega }{r}+m(r)&\mathrm {i}\partial _r +V(r)\end{bmatrix}\frac{1}{\sqrt{2}}\begin{bmatrix}1&-\mathrm {i}\\\mathrm {i}&-1\end{bmatrix}=\begin{bmatrix}-\frac{\omega }{r}+V(r) & - \partial _r -m(r)\\\partial _r -m(r)& \frac{\omega }{r} +V(r)\end{bmatrix}.\) 

For \(m=0\)  and \(V=-\frac{\lambda }{r}\) , this is the 1-dimensional Dirac operator studied in our paper.

Composite angular momentum
Introduce the spin operators
\(\sigma _{ij}&:=-\frac{\mathrm {i}}{2}[\alpha _i,\alpha _j] .\) 

\(\frac{1}{2}\sigma _{ij}\)  yield a representation of \(\mathfrak {so}(d)\)  on the spin space \(\mathcal {K}\) :
\(\Big [\frac{1}{2}\sigma _{ij},\alpha _k\Big ]&=-\mathrm {i}\delta _{jk}\alpha _i+\mathrm {i}\delta _{ik}\alpha _j,\\\Big [\frac{1}{2}\sigma _{ij},\frac{1}{2}\sigma _{kl}\Big ]&=-\mathrm {i}\delta _{jk}\frac{1}{2}\sigma _{il}-\mathrm {i}\delta _{il}\frac{1}{2}\sigma _{jk}+\mathrm {i}\delta _{ik}\frac{1}{2}\sigma _{jl}+\mathrm {i}\delta _{jl}\frac{1}{2}\sigma _{ik}. \) 


Its quadratic Casimir operator is
\(\frac{\sigma ^2}{4}=\frac{1}{4} \sum _{i<j}\sigma _{ij}^2=&\frac{d(d-1)}{8}.\) 
Irreducible representations of \(\mathfrak {so}(d)\) 
contained in \(\mathcal {K}\)  will be called spinor representations.
Quadratic Casimir in these representations is given by (REF ).
If \(d\)  is even, then there are two inequivalent spinor
representations. They correspond to the eigenspaces of \(\beta \)  with
eigenvalues \(\pm 1\) .
If \(d\)  is odd, then \(\mathcal {K}\)  is also a direct sum of two spinor
representations, however they are equivalent
to one another. The decomposition of \(\mathcal {K}\)  into irreducible components
exists but
is clearly non-unique. One possible choice
corresponds to the eigenvalues \(\pm 1\) 
of \(\beta \) .
We also have the composite representation of \(\mathfrak {so}(d)\)  given by
\(J_{ij}&:=L_{ij}+\frac{1}{2}\sigma _{ij}.\) 

Clearly,
\([J_{ij},J_{kl}]&=-\mathrm {i}\delta _{jk}J_{il}-\mathrm {i}\delta _{il}J_{jk}+\mathrm {i}\delta _{ik}J_{jl}+\mathrm {i}\delta _{jl}J_{ik},\\[J,x\cdot \alpha ]&=[J,p\cdot \alpha ]=[J,p^2] =[J,x^2]=0 . \) 


The quadratic Casimir of this representation, also called the square of the
total angular momentum, is
\(J^2=&\sum _{i<j}J_{ij}^2=L^2+L\sigma +\frac{\sigma ^2}{4},\\\text{where } \qquad L\sigma &:=\sum _{i<j}L_{ij}\sigma _{ij}.\) 
Proposition 55 We have the following relation:
\(\kappa ^2=J^2+\frac{(d-1)(d-2)}{8}.\) 

Directly from the definition we have
\(\kappa ^2 = \frac{(d-1)^2}{4} + (d-1) L \sigma + \sum _{\begin{array}{c}i < j \\ k < l\end{array}} L_{ij} L_{kl} \sigma _{ij} \sigma _{kl}.\) 
To simplify the last term we write
\(\sigma _{ij} \sigma _{kl} = \frac{1}{2} [\sigma _{ij}, \sigma _{kl}] + \frac{1}{2} [\sigma _{ij}, \sigma _{kl}]_+.\) 
A simple expression for the first term is given by (REF ). The second one is
\(\frac{1}{2} [\sigma _{ij}, \sigma _{kl}]_+ = - \alpha _{[ i} \alpha _j \alpha _k \alpha _{l]} + 2 \delta ^{[i}_{[k} \delta ^{j]}_{l]}\) 
in which \([ \cdots ]\)  denotes skew-symmetrization of the enclosed indices. In order to prove this formula, first note that both sides are skew-symmetric with respect to the transposition of \(i\)  and \(j\)  or \(k\)  and \(l\) , so we may assume that \(i \ne j\)  and \(k \ne l\) . We have three cases. If sets \(A = \lbrace  i , j \rbrace \)  and \(B=\lbrace  k , l \rbrace \)  are disjoint, then all \(\alpha \)  matrices involved anticommute and hence both sides are equal to \(- \alpha _i \alpha _j \alpha _k \alpha _l\) . If \(A \cap B\)  has one element, one checks that both sides vanish. Finally, if \(A=B\)  then both sides are equal to \(\pm 1\) , the sign depending on the order of indices.
Now plug (REF ) and (REF ) into (REF ). The term with \(\alpha _{[ i} \alpha _j \alpha _k \alpha _{l]}\)  drops out after summing over indices because \(L_{[ij} L_{kl]}=0\) . In the term with (REF ) we can replace \(L_{ij} L_{kl}\)  by \(\frac{1}{2} [L_{ij},L_{kl}]\) , by skew-symmetry with respect to \(ij \leftrightarrow kl\) . After simplifications with (REF ) and (REF ) we obtain the claim.
Recall that on \(\mathcal {H}_\omega \)  the operator \(\kappa \)  acts as multiplication by \(\omega \) . We will now characterize \(\mathcal {H}_\omega \)  more closely.
Proposition 56 
Let \(\omega \)  be such that \(\mathcal {H}_\omega \ne \lbrace  0 \rbrace \) . Then there exist \(\ell \)  and subspaces \(\mathcal {W}_\ell ,\mathcal {W}_{\ell -1}\subset L^2(\mathbb {R}^d)\)  spherical of degree
\(\ell \)  resp. \(\ell -1\)  such that
\(\mathcal {H}_{\omega }&\subset (\mathcal {W}_\ell {\oplus }\mathcal {W}_{\ell -1})\otimes \mathcal {K},\\|\omega | &= \ell + \frac{d-3}{2},  \\\left. J^2 \right|_{\mathcal {H}_\omega } &=\ell ^2+\ell (d-3)+\frac{d^2-9d+16}{8}. \) 

Exceptional cases \(d=1,2\)  are easy to analyze separately: one has \(\kappa =0\)  in the former case and \(\kappa = \pm L_{12} + \frac{1}{2} \beta \)  (with the sign depending on the choice of Clifford matrices) in the latter. From now on we assume that \(d \ge 3\) . We note that (REF ) and \(J^2 \ge 0\)  imply that \(\omega \ne 0\) .
\(\kappa \)  commutes with \(\beta \) , hence also with
\(L\sigma \) . Therefore we can decompose \(\mathcal {H}_\omega \) 
with respect to the eigenvalues of \(L\sigma \) . From (REF ) we obtain
\( L\sigma =& \omega \beta -\frac{d-1}{2},\) 

which has on \(\mathcal {H}_\omega \)  two distinct eigenvalues
\(\pm \omega -\frac{d-1}{2}.\) 
Both sings are realized because \(D\)  anticommutes with \(\beta \)  and preserves \(\mathcal {H}_{\omega }\) .
Clearly \(L^2=J^2-L\sigma -\frac{\sigma ^2}{4}\)  has on \(\mathcal {H}_\omega \)  two distinct
eigenvalues corresponding to (REF ). As seen from
(REF ), the representation of orbital angular momentum is uniquely determined by \(L^2\) .
Therefore, for some \(\ell _+,\ell _-\in \mathbb {N}\) , \(\ell _+ > \ell _-\) ,
\(\mathcal {H}_{\omega }\subset (\mathcal {W}_{\ell _+}{\oplus }\mathcal {W}_{\ell _-})\otimes \mathcal {K}.\) 
Comparing the identities
\(J^2&=\ell _\pm (\ell _\pm +d-2)\mp |\omega |-\frac{d-1}{2}+\frac{d(d-1)}{8},\\J^2&=\omega ^2-\frac{(d-1)(d-2)}{8},\) 

we obtain the equation
\(|\omega |(|\omega |\pm 1)=\Big (\ell _\pm +\frac{d-1}{2}\Big ) \Big (\ell _\pm +\frac{d-3}{2}\Big ).\) 
whose solutions take the form \( \ell _+ + \frac{d-3}{2} \in \lbrace  |\omega |, -|\omega | - 1 \rbrace \) , \(\ell _- + \frac{d-3}{2} \in \lbrace  |\omega |-1, -|\omega | \rbrace \) . In both cases the second solution has to be discarded because \(\ell _\pm + \frac{d-3}{2} \ge 0\) . Hence () holds and \(\ell _- = \ell _+ - 1\) . Then () is obtain by feeding () into (REF ).
\(\Box \)
We remark that the sign of \(\omega \)  can not be obtained from the
above calculation. Indeed,
the spectrum of \(\kappa \)  on
\(L^2(\mathbb {R}^d)\otimes \mathcal {K}\)  is always invariant wrt \(\omega \mapsto -\omega \) .
If \(d\)  is odd, then \(\prod \limits _{j=1}^d \alpha _j\)  commutes with
\(L\)  and \(\alpha _i\) , but anticommutes with \(\beta \)  and hence with
\(\kappa \) .
If \(d\)  is even, then \(\kappa \)  anticommutes with the parity operator
\(f(x_1,\dots ,x_d) \mapsto \beta \alpha _1 f(-x_1, \dots , x_d), \qquad f \in L^2(\mathbb {R}^d, \mathcal {K}).\) 
However, this operation does not preserve the type of angular momentum representation. Indeed, it anticommutes with \(\beta \)  and hence exchanges the two spinor representations.

Analysis in various dimensions
Let us review the lowest dimensions.
\(\bf d=1\) . There is no angular momentum and one has \(\omega =0\) .
\(\bf d=2\) . Unitary irreducible representations of \(\mathfrak {so}(2)\)  are enumerated by
spin values \(m\in \mathbb {R}\) . The corresponding quadratic Casimir is equal to
\(m^2\) .
There are two types \(\mathcal {K}_{\pm \frac{1}{2}}\)  of spinor representations, corresponding
to \(m=\pm \frac{1}{2}\) .
Spherical representations correspond to \(\ell \in \mathbb {Z}\) .
One convenient choice of Clifford representation is given by Pauli matrices: \(\alpha _1 = \sigma _1\) , \(\alpha _2 = \sigma _2 \) , \(\beta =\pm \sigma _3\) . Then \(\kappa = \pm J\)  and hence
\(\mathcal {H}_{\omega }= (\mathcal {W}_\ell {\otimes }\mathcal {K}_{-\frac{1}{2}})\oplus (\mathcal {W}_{\ell -1}{\otimes }\mathcal {K}_{\frac{1}{2}}).\) 

with \(\omega = \pm (\ell - \frac{1}{2}) \in \mathbb {Z}+ \frac{1}{2}\) . Sign in the relation between \(\omega \)  and total angular momentum depends on the choice of sign in \(\beta \) , but after fixing Clifford matrices it is one-to-one.
\(\bf d=3\) . Unitary irreducible representations of \(\mathfrak {so}(3)\)  are parametrized by
spin \(j \in \frac{1}{2}\mathbb {N}\)  or the quadratic Casimir
\(j(j+1)\) . All spinor representations have the spin \(\frac{1}{2}\) .
The representation on \(\mathcal {H}_\omega \) 
has spin \(\ell -\frac{1}{2}\) . We have \(\omega =\pm \ell \in \lbrace  \pm 1 , \pm 2 , \dots \rbrace \) , i.e. two distinct values of \(\omega \)  correspond to the same total spin.
\(\bf d=4\) .
We have \(\mathfrak {so}(4)\simeq \mathfrak {so}(3)\oplus \mathfrak {so}(3)\) . More explicitly,
\(J_1^\pm :=\frac{1}{2}(\pm J_{12}+ J_{34}),&\quad J_2^\pm :=\frac{1}{2}(\pm J_{13}+ J_{42}),\quad J_3^\pm :=\frac{1}{2}(\pm J_{14}+J_{23})\) 

span two algebras isomorphic to \(\mathfrak {so}(3)\)  and commuting with one
another. Let \((J^\pm )^2\)  be the corresponding quadratic Casimirs. We
have
\( J^2=2(J^+)^2+2(J^-)^2.\) 
Thus irreducible representations of \(\mathfrak {so}(4)\)  are parametrized by pairs of spins
\((j^+,j^-)\in (\frac{1}{2}\mathbb {N})^2\)  with the quadratic Casimir
\(2j^+(j^++1)+2j^-(j^-+1)\) . We have also the obvious analogs of (REF ) and (REF ) for \(L_{ij}\)  and
\(\frac{1}{2}\sigma _{ij}\) .
Representations of \(\mathfrak {so}(4)\)  on spherical
harmonics satisfy
\(L_{12}L_{34}+L_{13}L_{42}+L_{14}L_{23}=0.\) 
Therefore,
\((L^+)^2=(L^-)^2\) . Hence a spherical representation of degree \(\ell \) 
corresponds to the pair of spins \((\frac{\ell }{2},\frac{\ell }{2})\) 
with the quadratic Casimir
\(\ell (\ell +2)=2\frac{\ell }{2}\Big (\frac{\ell }{2}+1\Big )+2\frac{\ell }{2}\Big (\frac{\ell }{2}+1\Big )\) . Spinor representations are of types \((\frac{1}{2},0)\)  and \((0,\frac{1}{2})\) . They satisfy
\(\alpha _1 \alpha _2 \alpha _3 \alpha _4 \, \sigma _i^\pm = \mp \sigma _i^\pm , \qquad (\sigma ^\pm )^2 = \mp \frac{3}{2} \alpha _1 \alpha _2 \alpha _3 \alpha _4.\) 
Furthermore, we have \(\beta = \pm \alpha _1 \alpha _2 \alpha _3 \alpha _4\) , with the sign in this relation distinguishing Clifford representation. Using these relations we derive
\(\kappa = \mp 2 (J^+)^2 \pm 2 (J^-)^2.\) 
From spherical representations and spinor representation it is
possible to build total angular momentum representations of two types:
\((\frac{\ell }{2},\frac{\ell -1}{2})\)  and
\((\frac{\ell -1}{2},\frac{\ell }{2})\) . They have the same quadratic Casimir
\(J^2=\ell (\ell +1)-\frac{1}{2}\) 
but can be distinguished by \(\omega \) :
\(\omega = \mp \left( \ell + \frac{1}{2} \right) \qquad \text{and} \qquad \omega = \pm \left( \ell + \frac{1}{2} \right).\) 
The inclusion (REF ) may now be stated more precisely:
\(\mathcal {H}_\omega \cong \left( \frac{\ell }{2}, \frac{\ell -1}{2} \right) \subset &\Big (\frac{\ell -1}{2},\frac{\ell -1}{2}\Big ){\otimes }\Big (\frac{1}{2},0\Big )\oplus \Big (\frac{\ell }{2},\frac{\ell }{2}\Big ){\otimes }\Big (0,\frac{1}{2}\Big ), \qquad \text{for } \pm \omega <0 \\\mathcal {H}_\omega \cong \left( \frac{\ell -1}{2},\frac{\ell }{2} \right) \subset &\Big (\frac{\ell -1}{2},\frac{\ell -1}{2}\Big ){\otimes }\Big (0,\frac{1}{2}\Big )\oplus \Big (\frac{\ell }{2},\frac{\ell }{2}\Big ){\otimes }\Big (\frac{1}{2},0\Big ), \qquad \text{for } \pm \omega >0.\) 


As in dimension 2, the relation between the total angular momentum representation and \(\omega \) , taking valued in \(\lbrace  \pm \frac{3}{2}, \pm \frac{5}{2}, \dots \rbrace \) , is one-to-one after fixing Clifford matrices.
For general dimensions we label irreducible
representations as in [10]}.
 \(\bf d=2n+1, \, n \ge 2\) .
Irreducible representations
are in \(1-1\)  correspondence with labels \((a_1,\dots ,a_n)\in \mathbb {N}^n\) . Spherical harmonics
of degree \(\ell \) 
have type \((\ell ,0,\dots )\) , while spinor representations have type \((0,\dots ,1)\) .
Their tensor product decomposes as
\((\ell ,\dots ,0)\otimes (0,\dots ,1)&= (\ell ,\dots ,1)\oplus (\ell -1,\dots ,1), \qquad \ell \ge 1.\) 

Thus the only possible types of \(\mathcal {H}_{\omega }\)  are \(( \ell -1, \dots , 1)\) . This representation occurs as a subrepresentation only in two tensor products:
\( (\ell ,\dots ,0){\otimes }(0,\dots ,1), \qquad (\ell -1,\dots ,0){\otimes }(0,\dots ,1).\) 
We have \(|\omega |=\ell +n -1\) , thus \(\omega \)  takes values
\(\lbrace  \pm n ,\pm ( n+1), \dots \rbrace \) , with opposite \(\omega \)  corresponding to the same total angular momentum. In particular it is not possible to express \(\kappa \)  as a polynomial in \(J_{ij}\) .
 \(\bf d=2n, \, n \ge 3\) .
Types of irreducible representations
are parametrized by \((a_1,\dots ,a_n)\in \mathbb {N}^n\) . \(\ell \) th degree spherical harmonics are of type
\((\ell ,0,\dots )\) . Spinor representations are of two types:
\((0,\dots ,1,0)\) .
and \((0,\dots ,0,1)\) .
We have tensor products decompositions (\(\ell \ge 1\) ):
\((\ell ,\dots ,0)\otimes (0,\dots ,1,0)&= (\ell ,\dots ,1,0)\oplus (\ell -1,\dots ,0,1),\\(\ell ,\dots ,0)\otimes (0,\dots ,0,1)&= (\ell ,\dots ,0,1)\oplus (\ell -1,\dots ,1,0).\) 


It follows that \(\mathcal {H}_{\omega }\)  must be of the type \((\ell -1,\dots ,0,1)\)  or \((\ell -1,\dots ,1,0)\) . These two
representations have the same quadratic Casimir, however they are
exchanged by the parity operator (REF ). Hence they can be
distinguished by the sign of the following Casimir element, defined
as the \(n\) th wedge power of the 2-form \(J\) :
\(\bigwedge _{j=1}^n J := \frac{1}{2^n} \epsilon ^{i_1 \dots i_{2n}} J_{i_1 i_2} \cdots J_{i_{2n-1} i_{2n}}.\) 
Here \(\epsilon \)  is the Levi-Civita symbol.
We will show that (REF ) is actually proportional to \(\kappa \) .
Using the fact that skew-symmetrization of the product of two or more \(L_{ij}\)  vanishes and Clifford relations we derive
\(\bigwedge _{j=1}^n J = \frac{n (2n-2)!}{2^{2n-2}} \left( (-\mathrm {i})^n \alpha _1 \cdots \alpha _{2n} \right) \left( L \sigma + \frac{2n-1}{2} \right).\) 
A Clifford representation is determined up to isomorphism by
specifying the sign in the relation \(\beta = \pm (-\mathrm {i})^n \alpha _1 \cdots \alpha _{2n}\) . Then we have
\(\bigwedge _{j=1}^n J = \pm \frac{n(2n-2)!}{2^{2n-2}} \kappa .\) 
As in lower even dimensions, for fixed Clifford matrices angular
momentum types are in one-to-one correspondence with the values \(\omega \in \lbrace  \pm \left( n - \frac{1}{2} \right), \pm \left( n + \frac{1}{2} \right), \dots \rbrace \) .

Mellin transformation
For a Schwartz function \(b\)  on \(\mathbb {R}\)  we define its Fourier
transform by
\( (\mathfrak {F} b)(k)=\int _{- \infty }^{\infty } b(t)\mathrm {e}^{-\mathrm {i}k t}t̥. \) 
It is extended to the space of Schwartz distributions in the usual way. Restriction of \(\frac{\mathfrak {F}}{\sqrt{2 \pi }}\)  to \(L^2(\mathbb {R})\)  is a unitary operator.
An isomorphism \(\iota : C_c^{\infty }(\mathbb {R}) \rightarrow C_c^{\infty }(\mathbb {R}_+)\)  is
defined by \((\iota f)(x)=x^{- \frac{1}{2}} f(\ln (x))\) . Dualizing,
we extend it to an isomorphism between spaces of distributions on \(\mathbb {R}\) 
and on \(\mathbb {R}_+\) . Restriction of \(\iota \)  to \(L^2(\mathbb {R})\)  is a unitary
operator onto \(L^2(\mathbb {R}_+)\) . By Schwartz class functions and tempered distributions on \(\mathbb {R}_+\)  we shall mean smooth functions (respectively distributions) on \(\mathbb {R}_+\)  which correspond through \(\iota \)  to Schwartz class functions (respectively tempered distributions) on \(\mathbb {R}\) .
Mellin transform is defined as the composition \(\mathfrak {M} = \mathfrak {F} \iota ^{-1}\) . It is an isomorphism between spaces of tempered distributions on \(\mathbb {R}_+\)  and on \(\mathbb {R}\) . If \(f \in C_c^{\infty }(\mathbb {R}_+)\) , then
\((\mathfrak {M} f)(k) = \int _0^{\infty } f(x) x^{- \frac{1}{2} - \mathrm {i}k} x̥.\) 
Recall that \(J\)  and \(K\)  are defined in Subsection
REF . We note the following identities:
\(\mathfrak {M} Jf(k) = \mathfrak {M} f(- k), \\A = \mathfrak {M}^{-1} K \mathfrak {M}.\) 
The following lemma will be used in Proposition REF .
The Mellin transformation plays here a secondary role.
Lemma 57 
Suppose that \(f_{\epsilon }\)  is a family of tempered distributions on \(\mathbb {R}_+\)  with parameter \(\epsilon \in ]0,1]\)  satisfying the following conditions:

\(\mathfrak {M} f_{\epsilon } \in L^1_{\mathrm {loc}}(\mathbb {R})\) ,

there exists \(g \in L^1_{\mathrm {loc}}(\mathbb {R})\)  such that \(\mathfrak {M} f_{\epsilon } \rightarrow g\)  pointwise for \(\epsilon \rightarrow 0\) ,

there exist \(c, N \ge 0\)  independent of \(\epsilon \)  such that \(|\mathfrak {M} f_{\epsilon }(k)| \le c (1+k^2)^N\)  for almost every \(k\) .

Then there exists a tempered distribution \(f_0\)  on \(\mathbb {R}_+\)  such that \(f_\epsilon \rightarrow f_0\)  for \(\epsilon \rightarrow 0\)  in the sense of tempered distributions. Moreover, \(\mathfrak {M} f_0 = g\) .

Lemma 58 
Let \(b\)  be a tempered distribution whose Mellin transform is a Borel function. Put
\((B^{\mathrm {pre}}f)(x) = \int _0^{\infty } b(xk) f(k) k̥, \qquad f \in C_c^{\infty }(\mathbb {R}_+).\) 
Then \(B^{\mathrm {pre}}f\)  is in \(L^2(\mathbb {R}_+)\)  and \(B^{\mathrm {pre}}\)  is a closable operator on \(L^2(\mathbb {R}_+)\)  with closure
\(B = \mathfrak {M} b(A) J.\) 


Whittaker functions
In this appendix we review some properties of special functions used
in this paper.
In particular, we discuss Whittaker
functions, which play the central role in our paper. We follow the conventions of
[11]} and [12]}.
Confluent equation
Before discussing the Whittaker equation and its solutions, let us say
a few words about the
closely related confluent equation and the hypergeometric equation.
The confluent
equation has the form
\(\big (z\partial _z^2 + (c-z)\partial _z - a\big )v(z)=0.\) 
Let us list three of its standard solutions:
\({}_1F_1(a;c;z)&\qquad \text{characterized by }\sim 1 \text{ near }0;\\z^{1-c} {}_1F_1(a+1-c;2-c;z) &\qquad \text{characterized by }\sim z^{1-c} \text{ near }0;\\z^{-a}{}_2F_0(a,a+1-c;-;-z^{-1}) &\qquad \text{characterized by }\sim z^{-a} \text{ near }\infty .\) 


We note that the function \({}_1\mathbf {F}_1(a;c;z)= \frac{1}{\Gamma (c)} {}_1 F_{1}(a;c;z)\)  is holomorphic in all variables. The other two solutions are defined for \(z \notin ] - \infty , 0 ]\) ; thus \({}_2 F_0(a,b;-;z)\)  is defined on \([0 , \infty [\) .
We will also need the hypergeometric equation
\(\big (z(1-z)\partial _z^2+(c-(a+b+1)z)\partial _z-ab\big )v(z)=0.\) 
Among its 6 standard solutions, members of the famous Kummer's
table, let us list three:
\({}_2F_1(a,b;c;z)&\qquad \text{characterized by }\sim 1 \text{ near }0;\\z^{1-c} {}_2F_1(a+1-c,b-c+1;2-c;z) &\qquad \text{characterized by }\sim z^{1-c} \text{ near }0;\\{}_2F_1(a,b;a+b+1-c;1-z) &\qquad \text{characterized by }\sim 1 \text{ near }1. \) 


\({}_2 F_1(a,b;c;z)\)  may be defined by a power series convergent for
\(z\)  in the unit disc. It admits analytic continuation along any path
in \(\lbrace  1 \rbrace \) . To make it a single-valued function, it is customary to restrict its domain to
\(z \in [1, \infty [\) .
Then \({}_2\mathbf {F}_1(a,b;c;z) = \frac{1}{\Gamma (c)} {}_2 F_1(a,b;c;z)\)  is holomorphic in all variables. It satisfies
\({}_2\mathbf {F}_1 (a,b;0;z) = abz {}_2\mathbf {F}_1(a+1,b+1;2;z),\) 
We have the following identities valid for \(z\notin ]-\infty ,0]\) :
\(&\frac{\sin (\pi c)}{\pi } z^{-a}{}_2F_0(a,a+1-c;-;-z^{-1})  \\ =&\frac{ {}_1\mathbf {F}_1(a;c;z)}{\Gamma (a+1-c)}-\frac{ z^{1-c}{}_1 \mathbf {F}_1a+1-c;2-c;z)}{\Gamma (a)}, \\[2ex]& \frac{\sin (\pi c)}{\pi } {}_2\mathbf {F}_1(a,b;a+b+1-c;1-z)  \\ =&\frac{ {}_2\mathbf {F}_1(a,b;c;z)}{\Gamma (a-c+1)\Gamma (b-c+1)}-\frac{ z^{1-c}{}_2 \mathbf {F}_1(a-c+1,b-c+1;2-c;z)}{\Gamma (a)\Gamma (b)}.\) 

The former formula may be taken as an alternative definition of \({}_2 F_0\) .
Lemma 59 Let \(\epsilon >0\) , \(\mathrm {Re}(z) < \frac{1}{2}\) . Then
\({}_2 F _1 (a,b+ \lambda , c+ \lambda ; z) = (1-z)^{-a} + O(\lambda ^{-1}), \qquad |\arg (\lambda )| \le \pi - \epsilon , \ |\lambda | \rightarrow \infty .\) 

Assumption about \(\arg (\lambda )\)  guarantees that for \(|\lambda |\)  sufficiently large \(c + \lambda \notin - \mathbb {N}\) , so the left hand side of (REF ) is finite. We apply the Pfaff transformation:
\({}_2 F _1 (a,b+ \lambda , c+ \lambda ; z) = (1-z)^{-a} {}_2 F_1 \left( a,c-b ; c + \lambda ; \frac{z}{z-1} \right).\) 
The claim follows from the standard series defining \({}_2 F_1\) , because \(\left| \frac{z}{z-1} \right| < 1\) .
Lemma 60 The following asymptotic expansion holds for \(\mathrm {Re}(z) < \frac{1}{2}\) , \(z \notin ] - \infty , 0 ]\) , \(s \rightarrow \pm \infty \) :
\({}_2 \mathbf {F}_1(a,b-\mathrm {i}s; c ; 1-z) \sim \mathrm {sgn}(s) \cdot \left( \frac{(1-z)^{-a} ( \mathrm {i}s)^{-a}}{\Gamma (c-a)} + \frac{z^{c-a-b+ \mathrm {i}s} (1-z)^{b-c+ \mathrm {i}s} (- \mathrm {i}s)^{a-c}}{\Gamma (a)} \right)\) 
locally uniformly in \(a,b,c,z\) .

Using () we get
 
\(&\frac{\sin \left( \pi (a+b-c+1 - \mathrm {i}s) \right)}{\pi } {}_2 \mathbf {F}_1(a,b-\mathrm {i}s, c ; 1-z)  \\= & \frac{{}_2 \mathbf {F}_1(a,b-\mathrm {i}s;a+b-c+1 - \mathrm {i}s; z)}{\Gamma (c-b+ \mathrm {i}s) \Gamma (c-a)} - z^{c-a-b+ \mathrm {i}s} \frac{{}_2 \mathbf {F}_1(c-b + \mathrm {i}s,c-a; c-a-b+1 + \mathrm {i}s ; z)}{\Gamma (a) \Gamma (b - \mathrm {i}s)}. \nonumber \) 


Then (REF ) gives for large \(|s|\) :
 
\(&\frac{\sin \left( \pi (a+b-c+1 - \mathrm {i}s) \right)}{\pi } {}_2 \mathbf {F}_1(a,b-\mathrm {i}s;c ; z) \\\sim & \frac{(1-z)^{-a}}{\Gamma (c-b+ \mathrm {i}s) \Gamma (c-a) \Gamma (a+b-c+1- \mathrm {i}s)} - \frac{z^{c-a-b+ \mathrm {i}s} (1-z)^{b-c- \mathrm {i}s}}{\Gamma (a) \Gamma (b - \mathrm {i}s) \Gamma (c-a-b+1 + \mathrm {i}s)}. \nonumber \) 


Using \(\sin \left( \pi (a+b-c+1 - \mathrm {i}s) \right) \sim \frac{1}{2 \mathrm {i}} \mathrm {e}^{\pi |s| + \mathrm {i}\pi \, \mathrm {sgn}(s) (a+b-c+1)}\)  and Stirling's formula
\(\Gamma (z_0 + z ) \sim \sqrt{\frac{2 \pi }{z}} \mathrm {e}^{-z} z^{z+z_0}, \qquad |\arg (z)| \le \pi - \epsilon , \ |z| \rightarrow \infty \) 
yields, after algebraic manipulations, formula (REF ).

Hyperbolic-type Whittaker equation
The standard form of
the Whittaker equation is
\(\Big ( -\partial _{z}^2 + \Big ( m^2 - \frac{1}{4} \Big ) \frac{1}{z^2} - \frac{ \beta }{ z } +\frac{1}{4} \Big )g=0.\) 

In this section we briefly describe solutions of the Whittaker equation, following mostly [11]}, [12]}.
We will sometimes call (REF ) the hyperbolic-type Whittaker
equation, to distinguish it from the trigonometric-type Whittaker
equation, which differs by the sign in front of \(\frac{1}{4}\) .
There are two kinds of standard solutions to the Whittaker equation.
The function \(\mathcal {I}_{\beta ,m}\)  is defined by
\(\mathcal {I}_{\beta ,m}(z) = z^{\frac{1}{2} + m} {\mathrm {e}^{ \mp \frac{z}{2}} }{}_1 \mathbf {F}_1 \Big ( \frac{1}{2} + m \mp \beta ; 1 + 2m ; \pm z \Big ) .\) 
The standard domain of \(\mathcal {I}_{\beta , m}\)  is \(]- \infty , 0 ]\) . We have
\(\mathcal {I}_{-\beta ,m}(-z)=\mathrm {e}^{-\mathrm {i}\pi (\frac{1}{2}+m) \mathrm {sgn}(\mathrm {Im}(z))} \mathcal {I}_{\beta ,m}(z) \qquad \text{for } z \in \mathbb {R},  \\\overline{\mathcal {I}_{\overline{\beta }, \overline{\vphantom{\beta }m}}(\overline{z})}= \mathcal {I}_{\beta , m}(z). \) 

The case \(2m\in \mathbb {Z}\)  is called degenerate, and then
\(\mathcal {I}_{\beta ,-m}(z)= \Big (-\beta - m+\frac{1}{2}\Big )_{2m}\, \mathcal {I}_{\beta ,m}(z).\) 
The function \(\mathcal {K}_{\beta ,m}\)  is defined by
\(\mathcal {K}_{\beta ,m}(z) &:=z^\beta \mathrm {e}^{-\frac{z}{2}} {}_2F_0\Big (\frac{1}{2}+m-\beta ,\frac{1}{2}-m-\beta ;-;-z^{-1}\Big )  \\&=\frac{\pi }{\sin (2\pi m)}\Big (-\frac{\mathcal {I}_{\beta ,m}(z)}{\Gamma (\frac{1}{2}-m-\beta )}+\frac{\mathcal {I}_{\beta ,-m}(z)}{\Gamma (\frac{1}{2}+m-\beta )}\Big ). \nonumber \) 

It satisfies
\(\mathcal {K}_{\beta ,-m}(z)=\mathcal {K}_{\beta ,m}(z),  \\\overline{\mathcal {K}_{\overline{\beta }, \overline{\vphantom{\beta }m}}(\overline{z})} = \mathcal {K}_{\beta , m}(z). \) 
The Wronskian of \(\mathcal {I}_{\beta ,m}(\cdot )\)  and \(\mathcal {K}_{\beta ,m}(\cdot )\)  takes the form
\(\mathcal {W}(\mathcal {I}_{\beta ,m},\mathcal {K}_{\beta ,m}) = - \frac{ 1}{ \Gamma \Big ( \frac{1}{2} + m - \beta \Big ) }. \) 
If \(\frac{1}{2} + m - \beta \in - \mathbb {N}\) , then the Wronskian vanishes and we have
\(\mathcal {K}_{\beta ,m}(z) =& \mathrm {e}^{\mathrm {i}\pi (\frac{1}{2} + m - \beta )} \Gamma \left( \frac{1}{2} + m + \beta \right) \mathcal {I}_{\beta ,m}(z).\) 

In this case functions in (REF ) essentially coincide with Laguerre polynomials
\(\mathcal {I}_{\pm (\frac{1}{2}+m+n),m}(z) = \frac{n!\;\!z^{\frac{1}{2}+m}\mathrm {e}^{\mp \frac{z}{2}}}{\Gamma (1+2m+n)}L_n^{(2m)}(\pm z).\) 
Asymptotics of \(\mathcal {I}_{\beta ,m}\)  for small arguments are of the form
\(\mathcal {I}_{\beta ,m}(z) =\frac{ z^{\frac{1}{2}+m} }{ \Gamma ( 1 + 2m ) } \Big ( 1 - \frac{\beta }{ 1 + 2 m } z + O( z^2 ) \Big ) , \text{ if } m \ne -\frac{1}{2},-1,-\frac{3}{2},\dots ,  \quad z \rightarrow 0 ,\) 
The function \(\mathcal {K}_{\beta ,m}\)  satisfies, for \(z \rightarrow 0\) ,

\(\begin{array}{rll}\mathcal {K}_{\beta ,m}(z) =&z^{\frac{1}{2}} \Big ( \frac{ \Gamma ( -2m ) }{ \Gamma ( \frac{1}{2} - m - \beta ) } z^m + \frac{ \Gamma ( 2 m ) }{ \Gamma ( \frac{1}{2} + m - \beta ) } z^{-m} \Big ( 1 - \frac{ \beta }{ 1 - 2 m } z \Big ) \Big )&\vspace{2.84544pt} \\&\qquad \qquad \qquad + O( |z|^{\frac{3}{2} + \mathrm {Re}(m)} )+O(|z|^{\frac{5}{2}-\mathrm {Re}(m)}) &\text{ for } \mathrm {Re}(m)\in [0,1[ , \, m\ne 0,\frac{1}{2} \vspace{2.84544pt} \\\mathcal {K}_{\beta ,0}(z) =&- \frac{ z^{\frac{1}{2}} }{ \Gamma ( \frac{1}{2} - \beta ) } \big ( \mathrm {ln}(z) + \psi ( \frac{1}{2} - \beta ) + 2\lambda \big ) + O( |z|^{\frac{3}{2}}\ln (z) ) , &\text{ for } m = 0 , \, \beta \notin \frac{1}{2} + \mathbb {N} \vspace{2.84544pt} \\\mathcal {K}_{\beta ,0}(z) =&(\beta -\frac{1}{2})!(-1)^{\beta -\frac{1}{2}} z^{\frac{1}{2}} + O( |z|^{\frac{3}{2}} ) , &\text{ for } m = 0 , \, \beta \in \frac{1}{2} + \mathbb {N} \vspace{2.84544pt} \\\mathcal {K}_{\beta ,\frac{1}{2}}(z) =&\frac{ 1 }{ \Gamma ( - \beta ) } \big ( -\frac{1}{\beta }+z\ln (z)+ z(\psi (1-\beta )+2\lambda -1+\frac{1}{2\beta })\big )&\vspace{2.84544pt} \\&\qquad \qquad \qquad + O( z^2\ln (z) ) , &\text{ for } m = \frac{1}{2} , \beta \notin \mathbb {N} \vspace{2.84544pt} \\\mathcal {K}_{0,\frac{1}{2}}(z) =&1-\frac{z}{2}+O(z^2),&\text{ for }m=\frac{1}{2},\beta =0\vspace{2.84544pt} \\\mathcal {K}_{\beta ,\frac{1}{2}}(z) =&\beta ! (-1)^{\beta - 1} z + O( z^2 ) , &\text{ for } m = \frac{1}{2} , \beta \in \mathbb {N}^\times \vspace{2.84544pt} \\\mathcal {K}_{\beta ,m}(z) =& \frac{ \Gamma ( 2m ) }{ \Gamma ( \frac{1}{2} + m - \beta ) } z^{\frac{1}{2} - m} + O( |z|^{\frac{3}{2} - \mathrm {Re}(m) } ) , &\text{ for } \mathrm {Re}(m) \ge 1 .\end{array}\) 

Here \(\gamma \)  denotes Euler's constant and \(\psi \)  is the digamma function.
Asymptotics for \(|z| \rightarrow \infty \)  are given by (\(\epsilon > 0\) ):
\(\mathcal {K}_{\beta ,m}(z)& = z^{\beta }\mathrm {e}^{-\frac{z}{2}} \big ( 1 + O( z^{-1} ) \big ) , \quad |\mathrm {arg}(z)|\le \frac{3}{2}\pi -\epsilon , \\\mathcal {I}_{\beta , m }(z) & = \frac{z^{- \beta } \mathrm {e}^{\frac{z}{2}}}{\Gamma ( \frac{1}{2} + m - \beta )} \left( 1 + O (z^{-1}) \right), \quad |\mathrm {arg}(z)| \le \frac{\pi }{2} - \epsilon .\) 
The analytic continuations of \(\mathcal {K}\) , and more precisely the functions
\(z \mapsto \mathcal {K}_{-\beta ,m}(\mathrm {e}^{\pm \mathrm {i}\pi }z)\)  are also solutions of
the Whittaker Equation (REF ). One can define \(\mathcal {K}_{- \beta , m}(\mathrm {e}^{\pm i \pi } z )\)  as the unique holomorphic function of \(z \in ]- \infty , 0]\)  which coincides with \(\mathcal {K}_{-\beta ,m}(-z)\)  on \(\mp \) . Then
\(\mathcal {K}_{-\beta ,m}(\mathrm {e}^{\pm \mathrm {i}\pi }z)&=\frac{\pi }{\sin (2\pi m)}\Big ( -\frac{\mathrm {e}^{\pm \mathrm {i}\pi (\frac{1}{2}+m)}\mathcal {I}_{\beta ,m}(z)}{\Gamma (\frac{1}{2}-m+\beta )}+\frac{\mathrm {e}^{\pm \mathrm {i}\pi (\frac{1}{2}-m)}\mathcal {I}_{\beta ,-m}(z)}{\Gamma (\frac{1}{2}+m+\beta )}\Big ).\) 

\(\big ( \mathcal {K}_{\beta ,m}(z), \mathcal {K}_{- \beta ,m}(\mathrm {e}^{\pm \mathrm {i}\pi } z) \big )\)  are linearly independent pairs of functions. In particular, \( \mathcal {I}_{\beta ,m}(z) \)  can be expressed in terms of these functions:
\(\mathcal {I}_{\beta ,m}(z) =\mathrm {e}^{\pm \mathrm {i}\pi \beta }\Big (\frac{\mathrm {e}^{\mp \mathrm {i}\pi (m-\frac{1}{2})}\mathcal {K}_{\beta ,m}(z)}{\Gamma (\frac{1}{2}+m+\beta )} +\frac{\mathcal {K}_{-\beta ,m}(\mathrm {e}^{\pm \mathrm {i}\pi }z)}{\Gamma (\frac{1}{2}+m-\beta )}\Big ).\) 

Trigonometric-type Whittaker functions
It is convenient, in parallel to (REF ), to consider
the trigonometric-type Whittaker equation
\(\Big ( -\partial _{z}^2 + \Big ( m^2 - \frac{1}{4} \Big ) \frac{1}{z^2} - \frac{ \beta }{ z } - \frac{1}{4}\Big ) g(z)=0.\) 

It can be easily reduced to the hyperbolic-type Whittaker equation.
The function \(\mathcal {J}_{\beta ,m}\)  is defined by the formula
\(\mathcal {J}_{\beta ,m}(z)= \mathrm {e}^{\mp \frac{\mathrm {i}\pi }{2} (m+ \frac{1}{2})} \mathcal {I}_{\mp \mathrm {i}\beta ,m} (\mathrm {e}^{\pm \frac{\mathrm {i}\pi }{2}}z).\) 
It may also be described without invoking analytic continuations beyond the principal branch:
\(\mathcal {J}_{\beta ,m}(z) ={\left\lbrace \begin{array}{ll}\mathrm {e}^{\frac{\mathrm {i}\pi }{2} \left( \frac{1}{2}+m \right)} \mathcal {I}_{\mathrm {i}\beta , m}(- \mathrm {i}z ), & - \frac{\pi }{2} < \mathrm {arg}(z) < \pi , \\\mathrm {e}^{-\frac{\mathrm {i}\pi }{2} \left( \frac{1}{2}+m \right)} \mathcal {I}_{-\mathrm {i}\beta , m}(\mathrm {i}z ), & - \pi < \mathrm {arg}(z) < \frac{\pi }{2}.\end{array}\right.}\) 
The two expression agree for \(|\mathrm {arg}(z)|< \frac{\pi }{2}\) , by (REF ). Combined with () this implies
\(\overline{\mathcal {J}_{\overline{\beta }, \overline{\vphantom{\beta }m}}}(\overline{z}) = \mathcal {J}_{\beta , m}(z).\) 
We also have a pair of functions \(\mathcal {H}_{\beta ,m}^\pm \)  defined by
\(\mathcal {H}_{\beta ,m}^\pm (z) = \mathrm {e}^{\mp \mathrm {i}\frac{\pi }{2}\left( {\frac{1}{2}+m} \right)}\mathcal {K}_{\pm \mathrm {i}\beta ,m}( \mp \mathrm {i}z)\) 
initially for \(\mathrm {Re}(z) >0\)  and extended analytically to \(z \in ] - \infty , 0 ]\) . By (), they satisfy
\(\overline{\mathcal {H}^{\pm }_{\overline{\beta }, \overline{\vphantom{\beta }m}}(\overline{z})} = \mathcal {H}^{\mp }_{\beta , m}(z). \) 
The following connection formula holds:
\(\mathcal {J}_{\beta ,m}(z)= \mathrm {e}^{-\pi \beta }\Big (\frac{\mathcal {H}_{\beta ,m}^+(z)}{\Gamma \left( {\frac{1}{2}+m+\mathrm {i}\beta } \right)} + \frac{\mathcal {H}_{\beta ,m}^-(z)}{\Gamma \left( {\frac{1}{2}+m-\mathrm {i}\beta } \right)}\Big ).\) 
For the behaviour around \(\infty \) , we have for \(x>0\) , \(x\rightarrow \infty \)
\(\mathcal {H}_{\beta ,m}^\pm (x) \sim \mathrm {e}^{\mp \mathrm {i}\frac{\pi }{2}(\frac{1}{2}+m)}\mathrm {e}^{\frac{\pi \beta }{2}}x^{\pm \mathrm {i}\beta }\;\!\mathrm {e}^{\pm \mathrm {i}\frac{x}{2}}\big (1 + O(x^{-1})\big ).\) 

Recurrence relations
Whittaker functions satisfy several recurrence relations.
There are 6 basic ones, which we quote after Appendix A5 of
[12]}.
\(\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}-m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,m}(z)&= \Big (-\frac{1}{2}-m-\beta \Big )\mathcal {I}_{\beta +\frac{1}{2},m+\frac{1}{2}}(z),\\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}+m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,m}(z)&=\mathcal {I}_{\beta -\frac{1}{2},m-\frac{1}{2}}(z), \\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}+m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,m}(z)&=\mathcal {I}_{\beta +\frac{1}{2},m-\frac{1}{2}}(z), \\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}-m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,m}(z)&=\Big (\frac{1}{2}+m-\beta \Big )\mathcal {I}_{\beta -\frac{1}{2},m+\frac{1}{2}}(z), \\\Big (z\partial _z+\beta -\frac{z}{2}\Big )\mathcal {I}_{\beta ,m}(z)&=\Big (\frac{1}{2}+m+\beta \Big )\mathcal {I}_{\beta +1,m}(z),\\\Big (z\partial _z-\beta +\frac{z}{2}\Big )\mathcal {I}_{\beta ,m}(z)&=\Big (\frac{1}{2}+m-\beta \Big )\mathcal {I}_{\beta -1,m}(z);\) 
\(\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}-m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= -\mathcal {K}_{\beta +\frac{1}{2},m+\frac{1}{2}}(z), \\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}+m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= \Big (-\frac{1}{2}+m+\beta \Big ) \mathcal {K}_{\beta -\frac{1}{2},m-\frac{1}{2}}(z), \\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}+m}{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= -\mathcal {K}_{\beta +\frac{1}{2},m-\frac{1}{2}}(z), \\\Big (\sqrt{z}\partial _z+\frac{-\frac{1}{2}-m}{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= \Big (-\frac{1}{2}-m+\beta \Big )\mathcal {K}_{\beta -\frac{1}{2},m+\frac{1}{2}}(z), \\\Big (z\partial _z+\beta -\frac{z}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= -\mathcal {K}_{\beta +1,m}(z),\\\Big (z\partial _z-\beta +\frac{z}{2}\Big )\mathcal {K}_{\beta ,m}(z)&= \Big (\frac{1}{2}+m-\beta \Big ) \Big (\frac{1}{2}-m-\beta \Big )\mathcal {K}_{\beta -1,m}(z).\) 

They have an interesting algebraic interpretation—they correspond to
the roots of the Lie algebra of generalized symmetries of the heat
equation in 2 dimensions, see [16]} (where they are presented
using confluent functions, which as we know are equivalent to
Whittaker functions).
These recurrence relations involve 1st order differentiation, so it is
tempting to expect that they are closely related to the
Dirac-Coulomb Hamiltonian. It turns out, however, that the relationship
is not direct. By easy algebraic manipulations involving
(REF )-() and (REF )-(),
we derive an additional pair of
recurrence relations described in the following proposition.
In some sense, (REF ) and (REF ) are “lower order” than
(REF ) and (REF ).
In fact, in (REF ) and (REF ) the parameters \(\mu ,\beta \) 
appear only in 0th order terms, whereas in (REF ) and (REF )
the differential operator is multiplied by \(\mu \) .
Proposition 61
\(\Big (2\mu \partial _x+\frac{2\mu ^2}{x}-\beta \Big )\mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x)&=\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x),\\\Big (2\mu \partial _x-\frac{2\mu ^2}{x}+\beta \Big )\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x)&=(\mu ^2-\beta ^2)\mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x);\) 
\(\Big (2\mu \partial _x+\frac{2\mu ^2}{x}-\beta \Big )\mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x)&=-(\mu +\beta )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x),\\\Big (2\mu \partial _x-\frac{2\mu ^2}{x}+\beta \Big )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x)&=(-\mu +\beta )\mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x).\) 

We first rewrite (REF )-() and
(REF )-() as follows:
\(\Big (\sqrt{z}\partial _z-\frac{\mu }{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(z)&= \Big (-\mu -\beta \Big )\mathcal {I}_{\beta +\frac{1}{2},\mu }(z),\\\Big (\sqrt{z}\partial _z+\frac{\mu }{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,\mu +\frac{1}{2}}(z)&=\mathcal {I}_{\beta -\frac{1}{2},\mu }(z), \\\Big (\sqrt{z}\partial _z+\frac{\mu }{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,\mu +\frac{1}{2}}(z)&=\mathcal {I}_{\beta +\frac{1}{2},\mu }(z), \\\Big (\sqrt{z}\partial _z-\frac{\mu }{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(z)&=\Big (\mu -\beta \Big )\mathcal {I}_{\beta -\frac{1}{2},\mu }(z); \) 
\(\Big (\sqrt{z}\partial _z-\frac{\mu }{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(z)&= -\mathcal {K}_{\beta +\frac{1}{2},\mu }(z), \\\Big (\sqrt{z}\partial _z+\frac{\mu }{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,\mu +\frac{1}{2}}(z)&= \Big (\mu +\beta \Big ) \mathcal {K}_{\beta -\frac{1}{2},\mu }(z), \\\Big (\sqrt{z}\partial _z+\frac{\mu }{\sqrt{z}}-\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,\mu +\frac{1}{2}}(z)&= -\mathcal {K}_{\beta +\frac{1}{2},\mu }(z), \\\Big (\sqrt{z}\partial _z-\frac{\mu }{\sqrt{z}}+\frac{\sqrt{z}}{2}\Big )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(z)&= \Big (-\mu +\beta \Big )\mathcal {K}_{\beta -\frac{1}{2},\mu }(z). \) 
Then we compute
\(&\frac{1}{\sqrt{z}}\big (-(\ref {rei1.})+(-\mu +\beta )(\ref {rei2.})-(\mu +\beta )(\ref {rei3.})+(\ref {rei4.})\big ),\\&\frac{1}{\sqrt{z}}\big ((\mu -\beta )(\ref {rei1.})+(-\mu ^2+\beta ^2)(\ref {rei2.})+(\mu ^2-\beta ^2)(\ref {rei3.})+(\mu +\beta )(\ref {rei4.})\big ),\\[2ex]&\frac{1}{\sqrt{z}}\big (-(\mu +\beta )(\ref {rek1.})+(\mu -\beta )(\ref {rek2.})+(\mu +\beta )(\ref {rek3.})+(\mu +\beta )(\ref {rek4.})\big ),\\&\frac{1}{\sqrt{z}}\big ((\mu -\beta )(\ref {rek1.})+(\mu -\beta )(\ref {rek2.})-(\mu -\beta )(\ref {rek3.})+(\mu +\beta )(\ref {rek4.})\big ).\) 

\(\Box \)
(REF ) and (REF ) are closely related to the
Dirac-Coulomb Hamiltonian. To see this relation let
us introduce \(\omega \)  satisfying \(\omega ^2=\mu ^2-\beta ^2\) . Then
(REF ) and (REF ) can be rewritten in the following form:
\(0=&2\mu \partial _x\big (\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x)+\mathrm {i}\omega \mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x)\big )\\&+\Big (-\frac{2\mu ^2}{x}+\beta -\mathrm {i}\omega \Big )\big (\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x)-\mathrm {i}\omega \mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x)\big ),\\0=& 2\mu \partial _x\big (\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x)-\mathrm {i}\omega \mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x)\big )\\&+\Big (-\frac{2\mu ^2}{x}+\beta +\mathrm {i}\omega \Big )\big (\mathcal {I}_{\beta ,\mu -\frac{1}{2}}(x)+\mathrm {i}\omega \mathcal {I}_{\beta ,\mu +\frac{1}{2}}(x)\big );\\[3ex]0=&2\mu \partial _x\big ((\mu +\beta )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x)-\mathrm {i}\omega \mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x)\big )\\&+\Big (-\frac{2\mu ^2}{x}+\beta -\mathrm {i}\omega \Big )\big ((\mu +\beta )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x)+\mathrm {i}\omega \mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x)\big ),\\0=& 2\mu \partial _x\big ((\mu +\beta )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x)+\mathrm {i}\omega \mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x)\big )\\&+\Big (-\frac{2\mu ^2}{x}+\beta +\mathrm {i}\omega \Big )\big ((\mu +\beta )\mathcal {K}_{\beta ,\mu -\frac{1}{2}}(x)-\mathrm {i}\omega \mathcal {K}_{\beta ,\mu +\frac{1}{2}}(x)\big ).\) 

The eigenvalue equations for \(\xi _p^\pm \)  and \(\zeta _p^\pm \)  follow
directly from these identities.

Integral transforms
Let us compute a useful integral transform of the confluent function:
Lemma 62 Assuming \(\mathrm {Re}(b) >0\)  and \(|\mathrm {Re}(w)| < \mathrm {Re}(z)\) , one has
\(& \int _0^\infty x^{b-1}\mathrm {e}^{-zx} {}_1 \mathbf {F}_1(a;c;wx)x̥=z^{-b}\Gamma (b) {}_2 \mathbf {F}_1(a,b;c;z^{-1}w).\) 

Furthermore, if \(\mathrm {Re}(b) >0\) , \(\mathrm {Re}(b+1-c)>0\) , \(\mathrm {Re}(z)>0\) , \(w, z^{-1}w \notin ] - \infty , 0]\) , then
\( &\int _0^\infty x^{b-1}\mathrm {e}^{-zx}{}_2F_0\big (a,a+1-c;-;-(wx)^{-1}\big )(wx)^{-a}x̥ \\=&z^{-b} \Gamma (b)\Gamma (1+b-c) {}_2 \mathbf {F}_1(a,b;a+b+1-c;1-z^{-1}w).\) 

We expand the confluent function in a power series and integrate term
by term:
 
\(\int _0^\infty \sum _{n=0}^\infty \frac{(a)_n}{\Gamma (c+n)}\mathrm {e}^{-zx}x^{b+n-1}w^nx̥&=\sum _{n=0}^\infty \frac{\Gamma (b+n)(a)_n}{\Gamma (c+n)}\frac{w^n}{z^{n+b}}=z^{-b}\Gamma (b) \sum _{n=0}^\infty \frac{(a)_n(b)_n}{\Gamma (c+n)} \left( \frac{w}{z} \right)^n.\) 


This proves the first identity under additional assumption \(|w|<\mathrm {Re}(z)\) . The integrand can be
majorized by an integrable function for \(|\mathrm {Re}(w)|<\mathrm {Re}(z)\) . Therefore, we
can extend the identity by analytic continuation to this domain,
yielding (REF ).
(REF ), (REF ) and () and analytic
continuation imply (REF ).
\(\Box \)
The following identity, valid for \(v >0\) , \(\mathrm {Re}(\epsilon ) > 0\) , \(\mathrm {Re}(m+1 - \mathrm {i}s)>0\) , follows from (REF ):
\(& \int _0^{\infty } \mathrm {e}^{-\epsilon x} x^{- \frac{1}{2} - \mathrm {i}s} \mathcal {J}_{\beta , m} (vx) x̥  \\=& v^{m + \frac{1}{2}} \Big ( \epsilon \pm \mathrm {i}\frac{v}{2}\Big )^{-m -1 + \mathrm {i}s} \Gamma (m+1 - \mathrm {i}s) {}_2 \mathbf {F}_1 \left( m+ \frac{1}{2} \pm \mathrm {i}\beta , m+1 - \mathrm {i}s ; 2m+1 ; \frac{v}{\frac{v}{2}\mp \mathrm {i}\epsilon } \right). \) 
Proposition 63 
Let \(v>0\) , \(\mathrm {Re}(m) > -1\) . Then \(x\mapsto \mathrm {e}^{-0x}\mathcal {J}_{\beta ,m}(vx):= \lim \limits _{\epsilon \downarrow 0}\mathrm {e}^{-\epsilon x}\mathcal {J}_{\beta ,m}(vx)\)  is a tempered
distribution on \(\mathbb {R}_+\)  with the Mellin transform
\(& \int _0^{\infty } \mathrm {e}^{-0 x} x^{- \frac{1}{2} - \mathrm {i}s}\mathcal {J}_{\beta , m} (vx)x̥ \\=& v^{- \frac{1}{2}+\mathrm {i}s}2^{m+1-\mathrm {i}s} (\pm \mathrm {i})^{-m -1 + \mathrm {i}s} \Gamma (m+1 - \mathrm {i}s) {}_2 \mathbf {F} _1 \left( m+ \frac{1}{2} \pm \mathrm {i}\beta , m+1 - \mathrm {i}s ; 2m+1 ; 2\pm \mathrm {i}0\right), \) 

which is bounded by \(c_{\beta ,m} \left( \mathrm {e}^{- \frac{\pi }{2} (|s|+s)} |s|^{\mathrm {Im}(\beta )} + \mathrm {e}^{- \frac{\pi }{2} (|s|-s)} |s|^{- \mathrm {Im}(\beta )} \right)\) , with \(c_{\beta ,m}\)  a locally bounded function of \(\beta ,m\) .
Similarly, for any \(\mu \)  (including \(\mu =0\) ), \(x\mapsto \mathrm {e}^{-0x}\frac{1}{\mu }\big (\mathcal {J}_{\beta ,\mu -\frac{1}{2}}(vx)+ \beta \mathcal {J}_{\beta ,\mu +\frac{1}{2}}(vx)\big )\)  is a tempered
distribution on \(\mathbb {R}_+\) , whose Mellin transform can be computed
from (REF ) and is bounded by \(c_{\beta , \mu } \left( \mathrm {e}^{- \frac{\pi }{2} (|s|+s)} |s|^{\mathrm {Im}(\beta )} + \mathrm {e}^{- \frac{\pi }{2} (|s|-s)} |s|^{- \mathrm {Im}(\beta )} \right)\) , with \(c_{\beta ,\mu }\)  a locally bounded function of \(\beta ,\mu \) .

We use the criterion from Lemma REF . Let
\(f_{\epsilon }(x)=\mathrm {e}^{-\epsilon x}\mathcal {J}_{\beta ,m}(vx) \) . Then \(t \mapsto \mathrm {e}^{\frac{t}{2}} f_{\epsilon }(\mathrm {e}^t)\)  is smooth and vanishes exponentially for \(t \rightarrow - \infty \)  and superexponentially for \(t \rightarrow \infty \) . In particular \(f_{\epsilon }\)  is a tempered distribution on \(\mathbb {R}_+\) . Its Mellin transform is given by the absolutely convergent integral (REF ). It is a smooth function with smooth pointwise limit \(\epsilon \rightarrow 0\) . Required bounds follow from (REF ). This completes the proof of the first part.
Next, we compute

\(& \int _0^{\infty } \mathrm {e}^{-\epsilon x} x^{- \frac{1}{2} - \mathrm {i}s}\frac{1}{\mu } \left( \mathcal {J}_{\beta , \mu - \frac{1}{2}} (vx) + \beta \mathcal {J}_{\beta , \mu + \frac{1}{2}}(vx) \right) x̥ \\= & v^\mu \left( \epsilon \pm \mathrm {i}\frac{v}{2} \right)^{- \mu - \frac{1}{2} + \mathrm {i}s} \Gamma \left( \frac{1}{2} + \mu - \mathrm {i}s \right) \\\times & \frac{1}{\mu } \Bigg ( {}_2 \mathbf {F}_1 \left( \mu + \mathrm {i}\beta , \frac{1}{2}+ \mu - \mathrm {i}s; 2 \mu ; \frac{v}{\frac{v}{2}- \mathrm {i}\epsilon } \right) - \mathrm {i}\beta \left( \frac{1}{2} + \mu - \mathrm {i}s \right) \frac{v}{\frac{v}{2}- \mathrm {i}\epsilon } {}_2 \mathbf {F}_1 \left( 1+ \mu + \mathrm {i}\beta , \frac{3}{2}+ \mu - \mathrm {i}s; 2 \mu +2 ; \frac{v}{\frac{v}{2}- \mathrm {i}\epsilon } \right) \Bigg ). \) 


Expression in the last line is nonsingular for \(\mu \rightarrow 0\)  on the
account of (REF ). Bounds on the growth at infinity are derived as in the first case.
\(\Box \)
Acknowledgement.
The work of J.D.
was supported by National Science Center (Poland) under the
grant UMO-2019/35/B/ST1/01651.
